Structural Synthesis of Parallel Robots: Part 4: Other Topologies with Two and Three Degrees of Freedom [1 ed.] 978-94-007-2674-1, 978-94-007-2675-8

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Structural Synthesis of Parallel Robots

SOLID MECHANICS AND ITS APPLICATIONS Volume 183

Series Editor:

G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series The fundamental questions arising in mechanics are:Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For other titles published in this series, go to www.springer.com/series/6557

Grigore Gogu

Structural Synthesis of Parallel Robots Part 4: Other Topologies with Two and Three Degrees of Freedom

123

Grigore Gogu Institut Franc¸ais de M´ecanique Avanc´ee LaMI BP 265, Aubi`ere Cedex France [email protected]

ISSN 0925-0042 ISBN 978-94-007-2674-1 e-ISBN 978-94-007-2675-8 DOI 10.1007/978-94-007-2675-8 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2010935808 c Springer Science+Business Media B.V. 2012  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

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n i put e r om C

N C Ry I - n s t i u d e r c a h m o

be ri né qu t y C d a nt N e s

I n × n - n× J ia t J x r n m c o - b k t l or a ni pu e m bs h f k 1 t l o a ni pu e r m bs h f k 2 a ni t pul m e r bs h x of c a ML I - b o r t e i d u c a n q é M t e i sr g I ( o u p ) r e c s hR G a i n g M A S E L - a bL o r t e i s d e n c S i t a ur x i é M p o e t a i uq A o m d ' r y t b ( L o f s e n c S i a d t e M r a c t ) i fu A o m n d r ,

a t r i x m de n y

r or l c a h ni r - e g E i l q u e E r , oc n t ' a l i s f or l e c - E

i

X V I P arc e f

M R L I - r a e bL o i t a u e t , i n f q o r I m d ' d e i q u e o t b R e t

d e l r e c - oi é M i t q u e r o n d l p M ( a y bL o f s ) i t l c r e M on a d , b R

put e r m o C c S i -

m s e fd ba t i h x c ng l u k r o m M

i es c m h a n l r o f b p t y F

t hl i dw a c e s o n b m fk y - es c m h a n i l p r t b o f v u n ≡n Gi iescmhanlrtpogfvO 0 x 0 y 0 z 0 e r f c a m -n p - be r l t num a o i es c m h a n l r t p j f o p G i i n t b je r s o u f m P a t j pr i c n s m o P i a n t j o c s p r m u e -d * P i a nt j o dl c e s pr m aP - R | l op t r e a m g n y t e ol d r u v a c h i n p w m g y aP - R R | a* P or a P c s - S C | R j a o i l p h ce r d s n y b t m aP c - |R C t pe y - r a m l og e p l o p t h i w one l e i d i ob t l y m c om t j i l n c ar o d y b e a P s u - pa r e l m og l o p i w t h e t hr e i dl s obi e t l m c o s p h e ric a l a n d a u a P s - RS | pe t y - pa r l m og e l o p i w t h t o w i dl e obi e t s l m c t j i l n ce ar o s p h b d a P s - | RS pe t y - r a m p l og e l o p i w t h l e i d obi e t s l m e a m h t s o nj c d i l r p w b aP t - R ⊥ P ⊥ | R | ⊥ P a P c t s - r i t a cl m s e o g p h w d n t j c ar o i l s p h e d y u a P - pa r e l m og i w t h n d ob l y m c t j oi n a P u - pa r e l m og l o p i w t h t o w e i dl obi t s l e m b c om t j s oi le r n u a v t l or a ni p u e m M P n P o p e l a r s i b c t n y fd g m n* P or n P c s - e c l os p l o i w t h e on r e d g f o t y i ob l m a nd e t hr i ce r a s l p h n d b y o m t n P s - c l os e l o p h i w t one r d g f o bi t l y m a nd e t hr e s a o t m j h c nd i r l b p w n2 P obi t l y f m e s r h d g w p a n c cs n2* P or n2 P - e c l os p i w t h o e s r d g f o t l y i b m a nd e r t h i ce r a s l p h n d b y o m t M

F b mi

Gi

N

G i

F

F

F

G ←

1

-G 2

-… G k

,

G i b l i- m

t j oi n t p y e - r a m l og e p i w t h te hr l e i d ob - l m nt bi ne d m i n a n iv e rs a l jo in t - om - c om l i nk ⊥

|

R l op r ea m g t n c s i y s i b e t l n d c om ed n i a s lr u v i ne d i n t o w

e i dl nt j oi l a l e i d ob t s m k nl i l ei d nt j oi l a

P arc e f X I

i o b t l y rf e m s g h d p w a n c -

n3 P n3* P

or

cs

n3 P

- c l os e l o p i w t h r e t h e s r d g of obi l t m y a nd

l i a nc d r e y b t s o m h

t j o i l n p h ce ar d s

t pul or a ni e m M P q l a r t he i p n o s c d b u f m ɺq - ectorviljyn q i - he t n i f c ml a s p d r F - t o a l b e r n u m o f t j i e s ra m p h l o s e i t r

e c ha ni s m l i j i nt o e d ua h c e nc d p i e t h i es c m h a n l r b p d o F

r l - t o a l b e r n u m o f i n t j e s ra m p t h s e l o i t r h

e nc d p i e t h he bi n t d c om ps l

r

k bs i m l be r num f o t j i n o e t r s a m p t ha os t l r e i t h e p nd i

-

Gi

nc e i n t he

bi n e d c om ps l

G i b, l i m -

R ol j r ut i e v n R t j o i a e c nl r u v - d nt j i o dl e r u - v bus r l ho p m -

*R bR b* R

or

bR

sc

- pl a n r bus hom p l t h i w e r i dl s ob e t m c

d bi ne om

n t j c ar o i l s p h e d y R F

- t he c or v a c e s p f o e l i a v t r s oc e n t w b

i es c m h a n l r p t f o ( R F a c e s t p or ) h b f i v R G i - t h e c o r v a e s p f o e l i ar v t s o c t e n w b nc e r f s or t m pl f a i n t he a t i c ne m k a i n c h i es c m h a n l r t p ( R G i p a c re s t f o i v h b ) S st j p h e r o i l n c a S * o b i t s l e m d h n j p ar w c S F - t he y i c on v b t w he obi l m a nd he t f r i es c m h a n l r t p S G i - t he i c on y v b w e l m a nd t h r e f t he i a c n m k h F ← G 1 -G 2 -… G t pul or a ni e s m M P ST F e c h l a r m of p und y t g s a ni pul or t m e s M -T P U i e r n t j u s a o lv * U obi t l y e m d r h a j un s w v v , v 1 , v 2 , v 3 - e cs t or y v i l a n x ,y z t oi n s c ra p h e f d ɺx , y , z ɺ ɺ snatdiecorfv-m

he t obi l m d a n e t h F R

G ←

1

-G 2

F

e h obi l m a nd e t h G F

G ←

R

1

-G

1

-G

e dn c t o i s r o m f i

,

2

-… G k

2

-… G

k .

Gi

e n c r s f oa t m p l i n F ←

G

e r n c s t m p l f oa i k

G i t c e o nd i s r m f t h e l a r p es c m h n i .

H

i s ma n F

-… G k

,

X a r cP e f

α β , δ , r l ot e s a ng i α ɺ β , δ ɺ , ɺ s l e a on t g i r h f d v - m

ω , ωα , ωβ , ωδ ectorvuslaying-

0 i es c m h a n / t f k o x d b 1 ≡1 Gi echanislmprtfox–d 1 Gi - 2 Gi - n … Gi 1 A -2 A -… n A b s l i of m nk 1 B -2 B -… n B b s l i of m nk 1 C - 2 C n …C b s l i of m nk 1 D -2 D -n… D b l s i of m nk 1 a nd 2 i n t he a o pe b of t l y i m | - l pa r e n t pos i of nt j i o ; i c ons t r / e a d x r f o e s aP aP | l o ps r e a m g t h ⊥ - a pe r c ul ndi o t s f j i n s t ; e c o r / a d x t i a on P t he r a l m og e p l o p a r e r l i a pe c u nd t o he t di t j i o a c n pr s m | ⊥ i n t he i on t a a r c ul pe ndi t o e h i c on t r d f e t h d ua c s r m i p o nl t j u i e v r h f c d a | i n e t h ao ⊥ r a m l og e l o p a r e l i a r c u d pe n t o e t h t e ol r u v a x r e a l m p og nd t o he i s a x f t h t j oi n ⊥ ⊥ i n t he a o r a m l og e l o p a r e l i a r c u d pe n t o e t h t e ol r u v a x r e a l m p og l o p a nd l s o a t o he t s a xi of e t h t e d ua c -over j i l u t en o ⊥ ⊥ i n t he n t i a o r a m g l op t j s oi n R l op r a m g

F sb l i o f m n k G G

2

-… G k

,

2

G 3

G 4

-R1

- t he pa r l c ni s m t o w

aP

pe b of t y i m l nd a

P R

-G 1

1

G

2P

G ←

i

R aP pl e m xa t he not i a

t e sc i a nd e t h t f a c t ha t he e s a x of e t h ut e ol v r j o

s i nt f o a nd

aP

aP

s

l r ae p r f o pl e xa m t h no-

⊥ aP

e sc a t ndi t he c t f a t ha he t xe s a of e t ol r u v j i nt o

s in c on t r i e of e t h |

R ⊥ P ⊥

e h t i s a x f o t h e ar l i c n d y t j o i s p e r -

C-

t i c nj o a d r - p R ⊥ aP

|

⊥

t h e o l u v r s a x f t h e o n d s c a l p -r

aP -

e s of t he r f s i t a t c e d u ol v r R ⊥ aP

⊥

⊥

t h e o l r u v a x s f n d c e l p -r

aP -

e s of t he r f s i t

aP nd a l s o t e h a x s f o e t h ol u v r j s i nt of r pa aP

⊥ R|

⊥

⊥

aP -

t he e ol ut v r a xe s f o l a r o- e p

a r e c ul i p nd t o h a xe s f o t l r pa e v

aP ║R

s

ut e l o- a e

s

tc i n u o r d 1 I

This book represents Part 4 of a larger work on the structural synthesis of parallel robots. The originality of this work resides in combining new formulae for the structural parameters and the evolutionary morphology in a unified approach of structural synthesis giving interesting innovative solutions for parallel robots. Part 1 (Gogu 2008a) presented the methodology of structural synthesis and the systematisation of structural solutions of simple and complex limbs with two to six degrees of connectivity systematically generated by the structural synthesis approach. Part 2 (Gogu 2009a) presented structural solutions of translational parallel robotic manipulators (TPMs) with two and three degrees of mobility. Part 3 (Gogu 2010a) presented structural solutions of parallel robotic manipulators with planar motion of the moving platform. Part 4 of this work focuses on the structural solutions of other parallel robotic manipulators with two and three degrees of freedom. This section recalls the terminology, the new formulae for the main structural parameters of parallel robots (mobility, connectivity, redundancy and overconstraint) and the main features of the methodology of structural synthesis based on the evolutionary morphology presented in Part 1.

1 . r m i l o ng e T y Robots can be found today in the manufacturing industry, agricultural, military and domestic applications, space exploration, medicine, education, information and communication technologies, entertainment, etc. We have presented in Part 1 various definitions of the word robot and we have seen that it is mainly used to refer to a wide range of mechanical devices or mechanisms, the common feature of which is that they are all capable of movement and can be used to perform physical tasks. Robots take on many different forms, ranging from humanoid, which mimic the human form and mode of movement, to industrial, whose appearance is dictated by the function they are to perform. Robots can be categorized as robotic manipulators, wheeled robots, legged robots swimming robots, fly-

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 183, DOI 10.1007/978-94-007-2675-8_1, © Springer Science+Business Media B.V. 2012

2

1 Introduction

ing robots, androids and self reconfigurable robots which can apply themselves to a given task. This book focuses on parallel robotic manipulators which are the counterparts to the serial robots. The various definitions of robotics converge towards the integration of the design and the end use in the studies related to robotics. This book focuses on the conceptual design of parallel robots. Although the appearance and capabilities of robots vary greatly, all robots share the features of a mechanical, movable structure under some form of control. The structure of a robot is usually mostly mechanical and takes the form of a mechanism having as constituent elements the links connected by joints. i s n cam eth d k o j , . 1 L Serial or parallel kinematic chains are concatenated in the robot mechanism. The serial kinematic chain is formed by links connected sequentially by joints. Links are connected in series as well as in parallel making one or more closed-loops in a parallel mechanism. The mechanical architecture of parallel robots is based on parallel mechanisms in which a member called a moving platform is connected to a reference member by at least two limbs that can be simple or complex. The robot actuators are integrated in the limbs (also called legs) usually closed to the fixed member, also called the base or the fixed platform. The moving platform positions the robot end-effector in space and may have anything between two and six degrees of freedom. Usually, the number of actuators coincides with the degrees of freedom of the mobile platform, exceeding them only in the case of redundantly-actuated parallel robots. The paradigm of parallel robots is the hexapod-type robot, which has six degrees of freedom, but recently, the machine industry has discovered the potential applications of lower-mobility parallel robots with just 2, 3, 4 or 5 degrees of freedom. Indeed, the study of this type of parallel manipulator is very important. They exhibit interesting features when compared to hexapods, such as a simpler architecture, a simpler control system, highspeed performance, low manufacturing and operating costs. Furthermore, for several parallel manipulators with decoupled or uncoupled motions, the kinematic model can be easily solved to obtain algebraic expressions, which are well suited for implementation in optimum design problems. Parallel mechanisms can be considered a well-established solution for many different applications of manipulation, machining, guiding, testing, control, etc.

1.1 Terminology

3

The terminology used in this book is mainly established in accordance with the terminology adopted by the International Federation for the Promotion of Mechanism and Machine Science (IFToMM) and published in (Ionescu 2003). The main terms used in this book concerning kinematic pairs (joints), kinematic chains and robot kinematics are defined in Tables 1.1-1.3 in Part 1 of this work. They are completed by some complementary remarks, notations and symbols used in this book. IFToMM terminology (Ionescu 2003) defines a link as a mechanism element (component) carrying kinematic pairing elements and a joint is a physical realization of a kinematic pair. The pairing element represents the assembly of surfaces, lines or points of a solid body through which it may contact with another solid body. The kinematic pair is the mechanical model of the connection of two pairing elements having relative motion of a certain type and degree of freedom. In the standard terminology, a kinematic chain is an assembly of links (mechanism elements) and joints, and a mechanism is a kinematic chain in which one of its links is taken as a “frame”. In this definition, the “frame” is a mechanism element deemed to be fixed. In this book, we use the notion of reference element to define the “frame” element. The reference element can be fixed or may merely be deemed to be fixed with respect to other mobile elements. The fixed base is denoted in this book by 0. A mobile element in a kinematic chain G is denoted by nG (n=1, 2, …). Two or more links connected together in the same link such that no relative motion can occur between them are considered as one link. The identity symbol “≡” is used between the links to indicate that they are welded together in the same link. For example, the notation 1G≡0 is used to indicate that the first link 1G of the kinematic chain G is the fixed base. A kinematic chain G is denoted by the sequence of its links. The notation G (1G≡0-2G-…-nG) indicates a kinematic chain in which the first link is fixed and the notation G (1G-2G-…nG) a kinematic chain with no fixed link. We will use the notion of mechanism to qualify the whole mechanical system, and the notion of kinematic chain to qualify the sub-systems of a mechanism. So, in this book, the same assembly of links and joints G will be considered to be a kinematic chain when integrated as a sub-system in another assembly of links and joints and will be considered a mechanism when G represents the whole system. The systematization, the definitions and the formulae presented in this book are valuable for mechanisms and kinematic chains. We use the term mechanism element or link to name a component (member) of a mechanism. In this book, unless otherwise stated, we consider all links to be rigid. We distinguish the following types of links:

4

1 Introduction

a) monary link - a mechanism element connected in the kinematic chain by only one joint (a link which carries only one kinematic pairing element), b) binary link - a mechanism element connected in the kinematic chain by two joints (a link connected to two other links), c) polinary link - a mechanism element connected in the kinematic chain by more than two joints (ternary link - if the link is connected by three joints, quaternary link if the link is connected by four joints). The IFToMM terminology defines open/closed kinematic chains and mechanisms, but it does not introduce the notions of simple (elementary) and complex kinematic chains and mechanisms. A closed kinematic chain is a kinematic chain in which each link is connected with at least two other links, and an open kinematic chain is a kinematic chain in which there is at least one link which is connected in the kinematic chain by just one joint. In a simple open kinematic chain (open-loop mechanism) only monary and binary links are connected. In a complex kinematic chain at least one ternary link exists. We designate in each mechanism two extreme elements called reference element and final element. They are also called distal links. In an open kinematic chain, these elements are situated at the extremities of the chain. In a single-loop kinematic chain, the final element can be any element of the chain except the reference element. In a parallel mechanism, the two distal links are the moving and the reference platform. The two platforms are connected by at least two simple or complex kinematic chains called limbs. Each limb contains at least one joint. A simple limb is composed of a simple open kinematic chain in which the final element is the mobile platform. A complex limb is composed of a complex kinematic chain in which the final element is also the mobile platform. IFToMM terminology (Ionescu 2003) uses the term kinematic pair to define the mechanical model of the connection of links having relative motion of a certain type and degree of freedom. The word joint is used as a synonym for the kinematic pair and also to define the physical realization of a kinematic pair, including connection via intermediate mechanism elements. Both synonymous terms are used in this text. Usually, in parallel robots, lower pairs are used: revolute R, prismatic P, helical H , cylindrical C, spherical S and planar pair E. The definitions of these kinematic pairs are presented in Table 1.1 – Part 1. The graphical representations used in this book for the lower pairs are presented in Fig. 1.1(a)-(f). Universal joints and homokinetic joints are also currently used in the mechanical structure of the parallel robots to transmit the rotational motion between two shafts with intersecting axes. If the instantaneous velocities of the two shafts are always the same, the kinematic joint is homokinetic (from the Greek “homos” and “kinesis” meaning “same” and

1.1 Terminology

5

“movement”). We know that the universal joint (Cardan joint or Hooke’s joint) are heterokinetic joints. Various types of homokinetic joints (HJ) are known today: Tracta, Weiss, Bendix, Dunlop, Rzeppa, Birfield, Glaenzer, Thompson, Triplan, Tripode, UF (undercut-free) ball joint, AC (angular contact) ball joint, VL plunge ball joint, DO (double offset) plunge ball joint, AAR (angular adjusted roller), helical flexure U-joints, etc. (DudiŃǎ a n d c u o e is D 1 9 8 7 , DudiŃǎ et al. 1989, 2001a, b). The graphical representations used in this book for the universal homokinetic joints are presented in Fig. 1.1(g)-(h). Joints with idle mobilities are commonly used to reduce the number of overconstraints in a mechanism. The idle mobility is a potential mobility of a joint that is not used by the mechanism and does not influence mechanism’s mobility in the hypothesis of perfect manufacturing and assembling precision. In theoretical conditions, when no errors exist with respect to parallel, perpendicular or intersecting positions of joint axes, motion amplitude associated with an idle mobility is zero. Real life manufacturing and assembling processes introduce errors in the relative positions of the joint axes and, in this case, the idle mobilities become effective mobilities usually with small amplitudes, depending on the precision of the mechanism. For example, the idle mobilities which can be combined in the parallelogram loop in Fig. 1.2 are systematized in Table 1.1 along with the number r of parameters that lose their independence in the closed loop and the number of overconstraints N of the corresponding linkage. A joint can combine idle and non idle (effective) mobilities. A joint combining only idle mobilities is called idle joint.

Fig. 1.1. Symbols used to represent the lower kinematic pairs and the kinematic joints: (a) revolute pair, (b) prismatic pair, (c) helical pair, (d) cylindrical pair, (e) spherical pair, (f) planar contact pair, (g) universal joint, (h) homokinetic joint , (i) two superposed revolute joints (1-2) and (2-3) with the same axis, (j) superposed cylindrical (1-2) and revolute (2-3) joints with the same axis, (k) superposed revolute (1-2) and cylindrical (2-3) joints with the same axis, and (l) two superposed cylindrical joints (1-2) and (2-3) with the same axis

6

1 Introduction

Fig. 1.2. Parallelogram loops of types Pa (a), Pac (b), Pau (c), Pas (d), Pauu (e), Pacu (f), Pa* (g), Pasu (h) , Pass (i) and the number of r parameters that lost their independence in the closed loop

1.1 Terminology

7

Table 1.1. Parallelogram loops with idle mobilities and their corresponding number of overconstraints N No. Parallelogram loop 1 Pa (Fig. 1.2a) 2 Pac (Fig. 1.2b)

N

3

Pau (Fig. 1.2c)

2

4

Pas (Fig. 1.2d)

1

5

Pauu (Fig. 1.2e)

1

6

Pacu (Fig. 1.2f)

1

7

Pacs, Pa* (Fig. 1.2g) 0

8

Pasu (Fig. 1.2h)

0

9

Pass (Fig. 1.2i)

0

3 2

Idle mobilities No idle mobilities One translational idle mobility combined in a cylindrical joint One rotational idle mobility combined in a universal joint Two rotational idle mobilities combined in a spherical joint Two rotational idle mobilities combined in two universal joints One translational idle mobility combined in a cylindrical joint and one rotational idle mobilities combined in a universal joint One translational idle mobility combined in a cylindrical joint and two rotational idle mobilities combined in a spherical joint Three rotational idle mobilities combined in one revolute joint and one spherical joint Three idle mobilities combined in two spherical joints adjacent to the same link with a complementary internal rotational mobility of the link adjacent to the two spherical joints.

A parallel mechanism is a single or multi-loop linkage in which a moving link called characteristic link or platform is connected to a reference link (fixed base) by at least two non interconnected kinematic chains called limbs. A parallel robot can be illustrated by a physical implementation or by an abstract representation. The physical implementation is usually illustrated by robot photography and the abstract representation by a CAD model, structural diagram and structural graph. Figure 1.3 gives an example of the various representations of a Gough-Stewart type parallel robot largely used today in industrial applications. The physical implementation in Fig. 1.3a is a photograph of the parallel robot built by Deltalab (http://www.deltalab.fr/). In a CAD model (Fig. 1.3b) the links and the joints are represented as being as close as possible to the physical implementation (Fig. 1.3a). In a structural diagram (Fig. 1.3c) they are represented by simplified symbols, such as those introduced in Fig. 1.1, respecting the geometric relations defined by the relative positions of joint axes.

8

1 Introduction

Fig. 1.3. Various representations of a Gough-Stewart type parallel robot: physical implementation (a), CAD model (b), structural diagram (c) and its associated graph (d), A-limb (e) and its associated graph (f)

1.1 Terminology

9

A structural graph (Fig. 1.3d) is a network of vertices or nodes connected by edges or arcs with no geometric relations. The links are noted in the nodes and the joints on the edges. We can see that the GoughStewarttype parallel robot has six identical limbs denoted in Fig. 1.3c by A, B, C, D , E and F. The final link is the mobile platform 4≡4A≡4B≡4C≡4D≡4E≡4F and the reference member is the fixed platform 1A≡1B≡1C≡1D≡1E≡1F≡0. Each limb is connected to both platforms by spherical pairs. A prismatic pair is actuated in each limb. The spherical pairs are not actuated and are called passive pairs. The two platforms are polinary links, the other two links of each limb are binary links. The parallel mechanism 6-SPS-type associated with the Gough-Stewart type parallel robot is a complex mechanism with a multi-loop associated graph (Fig. 1.3d). It has six simple limbs of type SPS. The actuated pair is underlined. The simple open kinematic chain associated with A-limb is denoted by A (1A≡0-2A-3A-4A≡4) – Fig. 1.3e and its associated graph is tree-type (Fig. 1.3f). r b i t d o s h yl n a e , . 1 p 2 S We consider the general case of a robot in which the end-effector is connected to the reference link by k≥1 kinematic chains. The end-effector is a binary or polynary link called a mobile platform in the case of parallel robots, and a monary link for serial robots. The reference link may either be the fixed base or may be deemed to be fixed. The kinematic chains connecting the end-effector to the reference link can be simple or complex. They are called limbs or legs in the case of parallel robots. A serial robot can be considered to be a parallel robot with just one simple limb, and a hybrid robot a parallel robot with just one complex limb. We denote by F ← G1-G2-…-Gk the kinematic chain associated with a general serial, parallel or hybrid robot, and by Gi (1Gi-2Gi-…-nGi) the kinematic chain associated with the ith limb (i=1,2,…,k). The end effector is n≡nGi and the reference link 1≡1Gi. If the reference link is the fixed base, it is denoted by 1≡1Gi≡0. The total number of robot joints is denoted by p. A serial robot F ← G1 is a robot in which the end-effector n≡nG1 is connected to the reference link 1≡1G1 by just one simple open kinematic chain Gi (1Gi-2Gi-…nGi) called a serial kinematic chain. A parallel robot F ← G1-G2-…-Gk is a robot in which the end-effector n≡nGi is connected in parallel to the reference link 1≡1Gi by k≥2 kinematic chains Gi (1Gi-2Gi-…-nGi) called limbs or legs.

10

1 Introduction

A hybrid serial-parallel robot F ← G1 is a robot in which end-effector n≡nG1 is connected to reference link 1≡1G1 by just one complex kinematic chain G1 (1G1-2G1-…nG1) called complex limb or complex leg. A fully-parallel robot F ← G1-G2-…Gk is a parallel robot in which the number of limbs is equal to the robot mobility (k=M≥2), and just one actuator exist in each limb. A non fully-parallel robot F ← G1-G2-…Gk is a parallel robot with fewer number of limbs than the robot mobility (k0), to indicate the number of independent parameters in robot modelling and to determine the number of inputs needed to drive the mechanism. Earlier works on the mobility of mechanisms go back to the second half of the nineteenth century. During the twentieth century, sustained efforts were made to find general methods for the determination of the mobility of any rigid body mechanism. Various formulae and approaches were derived and presented in the literature. Contributions have continued to emerge in the last few years. Mobility calculation still remains a central subject in the theory of mechanisms. In Part 1 (Gogu 2008a) we have shown that the various methods proposed in the literature for mobility calculation of the closed loop mechanisms fall into two basic categories: a) approaches for mobility calculation based on setting up the kinematic constraint equations and calculating their rank for a given position of the mechanism with specific joint locations, b) formulae for a quick calculation of mobility without the need to develop the set of constraint equations. The approaches used for mobility calculation based on setting up the kinematic constraint equations and their rank calculation are valid without exception. The major drawback of these approaches is that the mobility cannot be determined quickly without setting up the kinematic model of the mechanism. Usually this model is expressed by the closure equations that must be analyzed for dependency. The information about mechanism mobility is derived by performing position, velocity or static analysis by using analytical tools (screw theory, linear algebra, affine geometry, Lie

12

1 Introduction

algebra, etc). For this reason, the real and practical value of these approaches is very limited in spite of their valuable theoretical foundations. Moreover, the rank of the constraint equations is calculated in a given position of the mechanism with specific joint locations. The mobility calculated in relation to a given configuration of the mechanism is an instantaneous mobility which can be different from the general mobility (global mobility, full-cycle mobility). The general mobility represents the minimum value of the instantaneous mobility in a free-of-singularity workspace. For a given mechanism, general mobility has a unique value for a free-of-singularity workspace. It is a global parameter characterizing the mechanism in all its configurations of the workspace except its singular ones. Instantaneous mobility is a local parameter characterizing the mechanism in a given configuration including singular ones. In a singular configuration the instantaneous mobility could be different from the general mobility. In this book, unless otherwise stated, general mobility is simply called mobility. Note 1. In a kinematotropic mechanism with branching singularities, full-cycle mobility is associated with each branch. In this case, the fullcycle mobility (global mobility) is replaced by the branch mobility which represents the minimum value of the instantaneous mobility inside the same free-of-singularity branch. As each branch has its own mobility, a single value for global mobility cannot be associated with the kinematotropic mechanisms (Gogu 2008b, 2009b, 2011a,b). The term kinematotropic was coined by K. Wohlhart (1996) to define the linkages that permanently change their full-cycle mobility when passing by an instantaneous singularity from one branch to another. Various single and multi-loop kinematotropic mechanisms have been presented in the literature (Wohlhart 1996, Dai and Jones 1999, Galletti and Fanghella 2001, Fanghella et al. 2006). A formula for quick calculation of mobility is an explicit relationship between the following structural parameters: the number of links and joints, the motion/constraint parameters of joints and of the mechanism. Usually, these structural parameters are easily determined by inspection without any need to develop the set of constraint equations. In Part 1, we have shown that several dozen approaches proposed in the last 150 years for the calculation of mechanism mobility can be reduced to the same original formula that we have called the Chebychev-GrüblerKutzbach (CGK) formula in its original or extended forms. These formulae have been critically reviewed (Gogu 2005b) and a criterion governing mechanisms to which this formula can be applied has been set up in (Gogu 2005c). We have explained why this well-known formula does not work for some multi-loop mechanisms. New formulae for quick calculation of

1.2 Methodology of structural synthesis

13

mobility have been proposed in (Gogu 2005d) and demonstrated via the theory of linear transformations. More details and a development of these contributions have been presented in Part 1. The connectivity between two links of a mechanism represents the number of independent finite and/or infinitesimal displacements allowed by the mechanism between the two links. The number of overconstraints of a mechanism is given by the difference between the maximum number of joint kinematic parameters that could lose their independence in the closed loops, and the number of joint kinematic parameters that actually lose their independence in the closed loops. The structural redundancy of a kinematic chain represents the difference between the mobility of the kinematic chain and connectivity between its distal links. Let us consider the case of the parallel mechanism F ← G1-G2-…-Gk in which the mobile platform n≡nGi is connected to the reference platform 1≡1Gi by k simple and/or complex kinematic chains Gi (1Gi-2Gi-…-nGi) called limbs. In Part 1, the following parameters have been associated with the parallel mechanism F ← G1-G2-…-Gk : RGi - the vector space of relative velocities between the mobile and the reference platforms, nGi and 1Gi, in the kinematic chain Gi disconnected from the parallel mechanism F, RF - the vector space of relative velocities between the mobile and the reference platforms, n≡nGi and 1≡1Gi, in the parallel mechanism F ← G1G2-…-Gk, whose basis is (RF)=( R

G1

∩R

G2

∩ . ∩ R

kG

),

(1.1)

SGi - the connectivity between the mobile and the reference platforms, nGi and 1Gi, in the kinematic chain Gi disconnected from the parallel mechanism F, SF - the connectivity between the mobile and the reference platforms, n≡nGi and 1≡1Gi, in the parallel mechanism F ← G1-G2-…Gk. We recall that the connectivity is defined by the number of independent motions between the mobile and the reference platforms. The notation 1≡1Gi≡0 is used when the reference platform is the fixed base. The vector spaces of relative velocities between the mobile and the reference platforms are also called operational velocity spaces. The following formulae demonstrated in Chapter 2-Part 1 (Gogu 2008a) for mobility MF, connectivity SF, number of overconstraints NF and redun-

14

1 Introduction

dancy TF of the parallel mechanism F ← G1-G2-…-Gk are used in structural synthesis of parallel robotic manipulators:

∑f p

M

= F

i

r−

(1.2)

, F

i =1

where S S F

= i m( d R ) i m( d R

NF=6q-rF ,

(1.3)

TF=MF-SF ,

(1.4)

= m( d i R )

iG

=R F

RG 1 ∩ )

.

∑S

∩

G2

∩

k

r F

=

− S

iG

(1.5)

,

iG

+ r F

l

kG

,

,

(1.6) (1.7)

i =1

∑p

,

k

p =

iG

(1.8)

i=1

and

q=p-m+1,

(1.9)

∑r

(1.10)

k

rl =

iG

.

i =1

We note that pGi represents the number of joints of Gi-limb, p the total number of joints of parallel mechanism F, m the total number of links in mechanism F including the moving and reference platforms, q the total number of independent closed loops in the sense of graph theory, fi the mobility of the ith joint, rF the total number of joint parameters that lose their independence in mechanism F, r iG the number of joint parameters that lose their independence in the closed loops of limb Gi, rl the total number of joint parameters that lose their independence in the closed loops that may exist in the limbs of mechanism F. In Eqs. (1.5) and (1.6), dim denotes the dimension of the vector spaces. We denote by k1 the number of simple limbs and by k2 the number of complex limbs (k=k1+k2). Eq. (1.8) indicates that the limbs of the parallel mechanism F ← G1-G2-…-Gk must be defined in such a way that a joint

1.2 Methodology of structural synthesis

15

must belong to just one limb; that is the same joint cannot be combined in two or more limbs. In Chapter 5-Part 1 the following structural conditions have been established: a) for the non redundant parallel robots (TF=0) SF=MF≤MGi

(i=1,…,k),

(1.11)

MGi=SGi≤6

(i=1,…,k),

(1.12)

SFSGi≤6

(i=1,…,k),

(1.14)

b) for the redundant parallel robots with TF>0

c) for the non overconstrained parallel robots (NF=0) MF= ∑ f i − 6 q p

,

(1.15)

i =1

d) for the overconstrained parallel robots with NF>0 MF> ∑ f i − 6 q p

.

(1.16)

i =1

We recall that

∑f p

M

iG

=

Gi

i

r−

iG

.

(1.17)

i=1

We note that the intersection in Eqs. (1.1) and (1.6) is consistent if the vector spaces RGi are defined by the velocities of the same point situated on the moving platform with respect to the same reference frame. This point is called the characteristic point, and denoted by H. It is the point with the most restrictive motions of the moving platform. The connectivity SF of the moving platform n≡nGi in the mechanism F ← G1-G2-…-Gk is less than or equal to the mobility MF of mechanism F. The basis of the vector space RF of relative velocities between the moving and reference platforms in the mechanism F ← G1-G2-…-Gk must be valid for any point of the moving platform n≡nGi. Note 2. The bases of vector spaces RGj and RF may contain up to 6 independent velocity vectors vx, vy, vz, ωα, ωβ and ωδ. We denote by vx, vy and vz the independent linear velocities of the characteristic point H of the moving platform and by ωα, ωβ and ωδ the independent angular velocities of

16

1 Introduction

the moving platform. For example the basis of vector space RGj of a planar limb with three revolute joint is always (RGj)=(vx, vy, ωδ) if the three joint axes are parallel to z0-axis. For the same dimension SGj, the basis of vector space RGj of certain kinematic chains may be defined by different combinations of velocity vectors vx, vy, vz, ωα, ωβ and ωδ. For example, in a spatial limb with three revolute joints with orthogonal axes and non zero distance between the joint axes adjacent to the same link, vector space RGj has always three dimensions, but the basis can be defined by various combination of three out of six vectors vx, vy, vz, ωα, ωβ and ωδ. In these cases, the bases of RGj in Eqs. (1.1) and (1.6) are selected such as the minimum value of S F is obtained by Eq. (1.6). By this choice, the result of Eq. (1.2) fits in with general mobility definition as the minimum value of the instantaneous mobility. In the same way, in certain parallel robots, for the same dimension SF, the basis of vector space RF may be defined by different combinations of velocity vectors vx, vy, vz, ωα, ωβ and ωδ. These solutions are called parallel robots with various combinations of rotational and translational velocities of the moving platform. In this case, the moving platform can make more than SF translational and/or rotational motions but just SF of them are independent motions. The parameters used in the new formulae (1.1)-(1.17) can be easily obtained by inspection with no need to calculate the rank of the homogeneous linear set of constraint equations associated with loop closure or with the rank of the complete screw system associated to the joints of the mechanism. An analytical method to compute these parameters has also been developed in Part 1 just for verification and for a better understanding of the meaning of these parameters. The following steps can be used for the calculation of structural parameters of a parallel mechanism based on formulae (1.1)-(1.10). Step 1: Identify the total number of links m (including the fixed base and the moving platform) and the total number of joints p in the parallel mechanism. Step 2: Calculate the number of independent closed loops q in the parallel mechanism, q=p-m+1. Step 3: Determine the number of limbs k connecting the moving platform to the fixed base such that no joint belongs to more than one limb, and check Eq. (1.8). Step 4: Identify the basis of RGj (j=1,2,…,k) by observing the independent motions between distal link nGj and 1Gj in the kinematic chain associated with Gj–limb disconnected from the parallel mechanism. Step 5: Calculate the connectivity between distal links nGj and 1Gj in the kinematic chain Gj disconnected from the parallel mechanism,

1.2 Methodology of structural synthesis

17

SGj=dim(RGj). If necessary, calculate the rank of the forward velocity Jacobian JGj of Gj-limb disconnected from the parallel mechanism, SGj=rank(JGj). Step 6: Calculate the connectivity between the distal links n≡nGj and 1≡1Gj in the parallel mechanism given by Eq. (1.6). Step 7: Determine the number of joint parameters that lose their independence in the closed loops that may exist in each limb. Step 8: Calculate the total number of joint parameters that lose their independence in the closed loops that may exist in the limbs of the parallel mechanism given by Eq. (1.10). Step 9: Calculate the total number of joint parameters that lose their independence in the parallel mechanism given by Eq. (1.7). Step 10: Calculate mobility MF, number of overconstraints NF and redundancy TF of the parallel mechanism given by Eqs. (1.2)-(1.4). These formulae have been successfully applied in Part 1 to structural analysis of various mechanisms including so called “paradoxical” mechanisms. These formulae are also useful for the structural synthesis of various types of parallel mechanisms with 2≤MF≤6 and various combinations of independent motions of the moving platform. These solutions are obtained in a systematic approach of structural synthesis by using the limbs generated by the method of evolutionary morphology presented in Part 1. h c ra po l g y m t u i n v . E1 2 Evolutionary morphology (EM) is a new method of systematic innovation in engineering design proposed by the author in (Gogu 2005a). EM is formalized by a 6-tuple of design objectives, protoelements (initial components), morphological operators, evolution criteria, morphologies and a termination criterion. The design objectives are the structural solutions, also called topologies, defined by the required values of mobility, connectivity overconstrained and redundancy and the level of motion coupling. The protoelements are the revolute and prismatic joints. The morphological operators are: (re)combination, mutation, migration and selection. These operators are deterministic and are applied at each generation of EM. At least MF=SF generations are necessary to evolve by successive combinations from the first generation of protoelements to a first solution satisfying the set of design objectives. Morphological migration could introduce new constituent elements formed by new joints or combinations of joints into the evolutionary process.

18

1 Introduction

Evolutionary morphology is a complementary method with respect to evolutionary algorithms that starts from a given initial population to obtain an optimum solution with respect to a fitness function. EM creates this initial population to enhance the chance of obtaining a “more global optimum”. Evolutionary algorithms are optimization oriented methods; EM is a conceptual design oriented method. A detailed presentation of the evolutionary morphology can be found in chapter 5 - Part 1. m t o i w c e p s h r b f l a y . T1 2 3

i g nl p u oc

Various levels of motion coupling have been introduced in Chapter 4 - Part 1 in relation with the Jacobian matrix of the robotic manipulator which is the matrix mapping (i) the actuated joint velocity space and the endeffector velocity space, and (ii) the static load on the end-effector and the actuated joint forces or torques. Five types of parallel robotic manipulators (PMs) are introduced in Part 1: (i) maximally regular PMs, if the Jacobian J is an identity matrix throughout the entire workspace, (ii) fully-isotropic PMs, if J is a diagonal matrix with identical diagonal elements throughout the entire workspace, (iii) PMs with uncoupled motions if J is a diagonal matrix with different diagonal elements, (iv) PMs with decoupled motions, if J is a triangular matrix and (v) PMs with coupled motions if J is neither a triangular nor a diagonal matrix. The term maximally regular parallel robot was recently coined by Merlet (2006a,b) to define isotropic robots. We use this term to define just the particular case of fully-isotropic PMs, when the Jacobian matrix is an identity matrix throughout the entire workspace. Isotropy of a robotic manipulator is related to the condition number of its Jacobian matrix, which can be calculated as the ratio of the largest and the smallest singular values. A robotic manipulator is fully-isotropic if its Jacobian matrix is isotropic throughout the entire workspace, i.e., the condition number of the Jacobian matrix is one. Thus, the condition number of the Jacobian matrix is an interesting performance index characterizing the distortion of a unit sphere under this linear mapping. The condition number of the Jacobian matrix was first used by Salisbury and Craig (1982) to design mechanical fingers and developed by Angeles (1997) as a kinetostatic performance index of the robotic mechanical systems. The isotropic design aims at ideal kinematic and dynamic performance of the manipulator (Fattah and Ghasemi 2002). In an isotropic configuration, the sensitivity of a manipulator is minimal with regard to both velocity and force errors and the manipulator can be controlled equally well in all direc-

1.2 Methodology of structural synthesis

19

tions. The concept of kinematic isotropy has been used as a criterion in the design of various parallel manipulators (Zanganeh and Angeles 1997; Tsai and Huang 2003). Fully-isotropic PMs give a one-to-one mapping between the actuated joint velocity space and the operational velocity space. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission. The various kinetostatic performance indices introduced in section 4.5-Part 1 have optimal values for fully-isotropic PMs (Gogu 2007f, 2008a, j). Table 1.2. Literature dedicated to maximally-regular and implicitly fully-isotropic parallel robotic manipulators No. Type of parallel robotic manipulator 1 T3-type

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

References

Carricato and Parenti-Castelli 2002 Gogu 2002, 2004a,d Gosselin and Kong 2002 Kim and Tsai 2002 Kong and Gosselin 2002a,b,c Rizk et al 2006 Stan et al. 2008 Wu et al. 2007 R2-type parallel wrist Gogu 2005f R3-type parallel wrist Gogu 2007b R3-type redundantly-actuated parallel Gogu 2007e wrists Planar T2R1-type Gogu 2004c, 2010b Spatial T2R1-type Gogu 2008g, 2009c, 2011a Zhang et al. 2009 Spatial T2R1-type with planar motion Gogu 2008i, 2010b of the moving platform T1R2-type Gogu 2005i T3R1-type with Schönflies Carricato 2005 motions Gogu 2004b, 2005g, 2006c, 2007a T2R2-type Gogu 2005h T1R3-type Gogu 2008h T1R3-type redundantly-actuated Gogu 2008e T2R3-type redundantly-actuated Gogu 2007d T3R2-type Gogu 2006b,d, 2009d T3R2-type redundantly-actuated Gogu 2006a T3R3-type hexapod Gogu 2006e

20

1 Introduction

The first solutions of maximally regular and implicitly fully-isotropic parallel robot were developed at the same time and independently by Carricato and Parenti-Castelli at University of Genoa, Kim and Tsai at University of California, Gosselin and Kong at University of Laval, and the author at the French Institute of Advanced Mechanics (IFMA). In 2002, the four groups published the first results of their works (Carricato and Parenti-Castelli 2002; Kim and Tsai 2002; Gosselin and Kong 2002; Kong and Gosselin 2002a, b; c; Gogu 2002). Each of the last three groups has built a prototype of this T3-type translational parallel robot in their research laboratories and has called this robot CPM (Kim and Tsai 2002), Orthogonal Tripteron (Gosselin et al. 2004) or Isoglide3-T3 (Gogu 2004a). The first physical implementation of this robot was the CPM developed at University of California by Kim and Tsai (2002). An innovative solution of fully-isotropic T3-type translational parallel robot called Pantopteron was recently proposed by Briot and Bonev (2009). In this solution based on pantograph linkages, the moving platform moves several times faster than its linear actuators. Various other types of maximally regular and implicitly fully-isotropic parallel robotic manipulators have been proposed in the last years (see Table 1.2). These solutions can be applied in machining applications (Gogu 2007c) or haptic devices (Gogu 2008f). Overconstrained and non overconstrained solutions of parallel manipulators with coupled, decoupled and uncoupled motions of the moving platform along with maximally regular solutions are presented in the following sections of this book. These solutions are actuated by linear and/or rotating motors situated on the fixed base or on a moving link. They have two or three degrees of mobility and combine various translational and rotational motions of the moving platform. Basic and derived solutions are presented in each chapter. No idle mobilities exist in the basic solutions. To reduce the number of overconstraints in the parallel robot, derived solutions are used. They are obtained from the basic solutions by combining various idle mobilities in the kinematic pairs. The following structural parameters associated with each solution presented in this book are systematized in the various tables of each chapter: m - number of links including the fixed base, pGi - number of joints in the Gi-limb, p - total number of joints in the parallel mechanism given by Eq. (1.8), q - number of independent closed loops in the parallel mechanism given by Eq. (1.9), k1 - number of simple limbs, k2 - number of complex limbs,

1.2 Methodology of structural synthesis

21

k - total number of limbs k=k1+k2, (RGi) - basis of the vector space of relative velocities between the moving and reference platforms in Gi-limb disconnected from the parallel mechanism, SGi - connectivity between the moving and reference platforms in Gilimb disconnected from the parallel mechanism, given by Eq. (1.5), rGi - number of joint parameters that lost their independence in the closed loops combined in Gi-limb, MGi - mobility of Gi-limb, given by Eq. (1.17), (RF) - basis of the vector space of relative velocities between the moving and reference platforms in the parallel mechanism given by Eq. (1.1), SF - connectivity between the mobile and reference platforms in the parallel mechanism given by Eq (1.6), rl - total number of joint parameters that lose their independence in the closed loops combined in the k limbs given by Eq. (1.10), rF - total number of joint parameters that lose their independence in the closed loops combined in the parallel mechanism given by Eq. (1.7), MF - mobility of the parallel mechanism given by Eq. (1.2), NF - number of overconstraint in the parallel mechanism given by Eq. (1.3), TF - degree of structural redundancy of the parallel mechanism given by Eq. (1.4), p ∑ j = 1 f j - total number of degrees of mobility in the pi joints of limb i, i

where fj is the mobility of joint j, p ∑ j = 1 f j - total number of degrees of mobility in the p joints of the par-

allel mechanism

l r f i d t n o m ac w s h y e 2 P m l r i g p ta f o n e v h

T he l r a p s i e c m h n w t a r e l f u s n i o s t c p l a h r e ui q ng o t s p a n i s . x e T y h a k m e b lp o s i n e t p d i l r a n s e i n t a do r p ( i s .a o n t x r h e c w d l

f o t he i ng ov m pl ar t f a nd t i e g or a body n

al r i c y nd o t m

on t i (

s-

T he l a e p r s e c m ha ni i w t h a l i c ndr y ot i n m of

t he i ng ov m pl a t -

f or m e ha v bi l t y s r t m p l f oa . i n p l o av r t m g f h e d c

M

he r w e l t s oc i v f ng , h p ar m s i a nd t j o e

a nd i t e y c v o w b h i n ov g m a d xe f i

2= F

. We l ca em h t

S F 2=

T h e i s b a f o e t h l a ir o n p e t c y v o r a c e s p ( R F ) =( e s od l a t b ci m n r k h , I

e n i v g by q. E ( 1 ) i s v ω 

1

=x

ω

a nd

α

α=

 = 

1

α

[ J]

q 1  q   2 

J

( 2 .1 )

a r e t he nd t e p i l a on t s r a nd l o rt i q

a r e t i h l s oc v f u2

ia t x . r J n m o b c h e s

2 ×2

P

pe t y - l r a s i c m h n w d y o -

T1 R R or

P i wt h R

M G

=S G

=2

a nd (

R F ) =(

v 1 , ωα ). Tehs

n o o f a c t i r se a h H s . a t xi o he n r d u

T o t n a i b no t da r e u

T1 R R or

pl e x c om b i w t h 2 t he s a i b of t s i l ona e r p . s c I t hi s , a y w e d. a i obt n c

2 ×2

q 1 a nd

T he s i c ba m l of

P

pe t y - l a r s e c h ni m t w l i c n- y

1 RT

v 1 , ωα ).

v

t i on e a r of pe s t y bs l i m e a k m a i l e v b e on i on rt a a nd ne o t l i a ns r t poi n

) d a n o e i n-

1T

) i o fn g t . v h e p l T a r m c d

1R

R

pe t y - a r l , e c h ni s m a b s i c l m

P t p e a P r . y - ( ) i F g 6 5h c s 1 d w l o

pl e on r s i m ≤ M

Gi

=S

Gi

≤ 6 t ha bi ne s om c l t v

v 1 a nd

ωα in

e i w d b r num of s i l n ut

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 183, DOI 10.1007/978-94-007-2675-8_2, © Springer Science+Business Media B.V. 2012

2 4 P e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

2 .1 l i r d n c a mo t y

1 TR

e p a t r - l yw m s i n u o c h d

n I t he 1 RT t o w n a t l ir o p e s i e l t o c v e n d p n o t h b o e d u a t c v 1 = v ( q 1 ,q ) 1

pe t y s l - a r i c m h n w

, t he

l i e cd a n t o r u p y m t j oi n : t s c i e l o v

ω

a nd 2

=ω α

( q ,q 1 ) α

. 2

i s n l o t u r ad e c . O 2 1 v T he e d c o u p l l i c d ar n y o t i n m a n d

∑

of

e o r vn cs ta i d l s i u o t n p 1

f i < 2 + 6q

.

s i ol S n ut of pe t y F

a n d a r i n e l r o n g t i a r o a s t c o r u n c a b e t er d . a n g bs l i m w t h R P . A l i c n d ar y t j o of e t h ol r u v d a n t i c s pr m j . n o e O r o h b t a pl e x a c t i ne m k i ns c ha i a ng t c o a t s lt e a e on a n pl p l e x a m s . i F g 4 1 0 n d 6 t a r P 1 ) . cE h n p l p l e , t h s a m o r . F i x n fc v d g pl e s i m d a n pl e x c om b i w t h a r ne l o u s c i . ( i F g 2 .1 a ) o r o n a i n g o v m l i n k . ( i F g 2 .1 b ) T h e s i P SU . ( i F g 2 .1 a ) P ⊥ R ⊥ t j . oi n e T h r bs l m f : i d ons t t he l i r ow ng a f u s c : m p ( R G1 ) =( nA

=S

Gi

Gi

-G 1

-G 2

i w t h a s r i v o u s r e d g f o t a ei r n c o s v 3

a nd t l a i d o e u c n b m i of pe s t y

=6

R or P

C a n c s o l be u d i n t h e a c b l m d i ns

a t e d u c bs l i m a r e - c om r e c d l os p l o e ( s r f o os e d l c p u t r i n .2 1 us e i la n F o gt e d a t u on h i xe fd s ba pl e m b l i m i s of pe s t y ro

S . ( i F g 2 .1 b ) n d a e t h p l e x m c o b l i m o f p e t y

PU

. i g ( F 2 1a nd ) b he T t c l u m s j o

Pa S

l e ar i c n d y t n d e l p io s u a m f h c -

q=2 S R

F

)= (

G2 F

a nd (

=2

=3

a nd

∩ R G3 ) =( M G1 =S v 1 v , 2 v , 3 , ω, α ω, β δω ), ( R G3 )=(

R F ) =(

T F =0

t i a e l nr b y o m

s t i on i n . i F g 2. T s hi o i b t l m y i s e t h a l t i on r 2. a ) r ound e t s h g b xi p y c f c e n t o i s t h . n k l e T h nt a l i r o b y m i s p l e t c o m .2 b i g n t s F ol u T he i l ons ut i n . i g F 2. e ha v t he ng l i f ow t s c r u M G1 =6, M M G2 , =7 ( v 1 v , 2 v , 3 , ω, α ω, β δω ), ( ) i ob e t s l a r m xn d w

G ←

e Ty h us e t o w d a t c e u M

,N

S F =2

pe t y - l e pa r s e c m ha ni i w t h

T1 R q t de n p i s l o p e t m t he i on t c d

R

∩ R

G1

G2

G1

=6

v 1 , ωα ). thoB nsotilu ehav , M G2 =S G2 =6 v 1 , ωα ), ( R F )=(

, M G3 =S v 1 , ωα ),

,

=2

G3

M

= F

. s t e xi n b l m G 2

of pe s t y

P S d an ot i n m of i nk l

SP

S i n t he l u- s o 3

. ( iF g B

a l i t o s j p h ce w r n - d d e by s l i nk

2 B - 3 B i n t he

l u r a : r e st a m p =2,

R

G3 G3

S

=S

G1

v 1 , ωα ), (

()= ,S F

=2

,N F

G2

R F ) =( =3

a nd

=6

,

S

v 1 , ωα ), T F =1

1) Tl . e 2 a b( s

G3

,

=2 M F

=3

( R e ( on l e i r nta

G1

()= R

G2

)=

2 p . T 1 e ya t r R c l - h s m i n o w u d

F gi . 2 1 t i hw c o p e u l d n r i a y o t m d r e h g s o a d c n o i t l eu b m

d n r co i a l t m 2 5

P a r e l c h s m i n o f p et y s

P RPa S -

( a ) dn a

US - C f oe r v c t sa ni b g m n

P RPa S -U

S -C

( b )

2 6 P e c a r n l h smi t w d y o

F gi . 2 . t i hw c o p e u l d i l c d n r a y o t i m n d a n t r e h d r e g s o etor ni a l b d m y c u

o n e h t f i vm p g a r l

P r a e l e c a h s m i n o f p et s y

P RPa S -

( a ) dn a f o e v r t c s a n i c o b n im g

S -C il b m

P RPa S -

S -C

( b )

2 p . T 1 e ya t r R c l - h s m i n o w u d

d n r co i a l t m 2 7 r c a u t l S p e m os h f i n

a Te l b 2 . 1

i F. n s 2 g 1 d a

c rt S u a l

No .

oit S nul P RPa S . ( g 2i F 1 a ) P RPa S -U . ( g 2i F 1 b ) 10 7 4 1 12 3 2

e t rp a m

1 m 2 p

1

3 p

2

4 p

3

5 p 6

q 7 k

8

1

k 9

R

1

( R

12 13 14 15 16 17 18 19 20 21 2 23 24 25 26 27 28 29

( R

30 31 32 m bu n em r o i l f c k s d t g h x a , o t b u a l n e m r j f is p h c o l e p h t n is a r c m

G2

)

G3

) 6 6

S

G3

(

3

( v 1 v , 2 v , 3 , ω, α ω, β δ ω )

0

0

G3

0 G1

6

G2

6

7

G3

2

3 15

F

15

2

3 3

F

3

0

F

1

f j

p

f j

f j

f j

2

p

j =1 3

j =1 p j =1

v 1 , ωα )

2 3

j =1

(

2

( v 1 , ωα ) 2

p

v 1 , ωα )

0

6

∑ ∑ ∑ ∑

(

2

G2

F

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

3

M

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

6

F

T

3 2

2

l

N

1 1

3

S r

3

6

R F) r

7

G1

r

S -C

9

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( v 1 , ωα )

G2

r

M

)

G1

r

M

G1

S

S

M

- CS

S -C

1 3

(

P RPa S . ( g 2i F a ) P RPa S . ( g 2i F b )

1 2

k 10

- UC S

1

9

9 6

7 2

2 17

18

b

p , k 1 b ne r u o m p f l i s ,

Gi

b u n e m r o j f is t h a

G i l b - ,i m , q bu n em r d o i p f t c l s k 2 b e r u o nc m f -

p

2 8 P e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

b p i l e x , sm o l e c i t bs n w h a g v m d r f p e p h t a r c l n smi , o is r f n m e t rp a o l m h s i d n c M Gi o b t i l m y f e t n o w h i a g vm d r f c p s l ti v b y e n o w h l a m d r c f p s o t bu a l n em r j f p i h s b i c eo d n m t h p d c e nt i hol s b m a r e p h t a r c l n smi T F d e r g c t o su f a l n ph y m j . to n i a b

f

j

=

Gi

h i

k

p

i =1

j

T

∑

(R

R

p

),

Gi G i

j =1

, N

(R

=6q r − F S−

∩ ( R

Gk

,

Gi

− S

Gi

i

i=1

=M

, =1 i , 2 . k

iG

) ∩ . F ),

G1

∑ r = ∑ S = ∑f

F F

f

o t bu a l n em r j f p i s h F

i

e c l a h smi n

=1 i , . 2 k

f j − r

p

F

N l

, r

,

2

i =1

M k

+k

i =1

F

bu n em r o j f t i b e, S F

F

F

+ r F

−r F

, .

, l

,

c o ne g

, r l

d e p n c h to l i s

,

Gi

k

r

Gi

G i i l b - ,m

F

b e r u o n m f vt c s a i h p l a c h si n m

k

r l=

, r

, ( R F ) b os i a t f e c v h r p l

h

k b i l sm

= d mi

Gi

d

e p h t n a r c l smi

( R F ) =( S F = d mi g

Gi

,

M

G i b il d - m c o s ne t r f

e c t on i v b y w h m d a g r f p l-

e a l i n c h sm

1

S e

) b os i a t f e c v h r p l -

e d o l b p i cs n m e

G i il b - m

k= d

Gi

G i i l b - m d c o s ne t r phf a

q =p - m +1 c

,( R

in S

∑

p=

c

k o t bu a l n em r f s i

),

l

, f j o b til m y f

de ,M F

o bm t i l y f k

j th

,

2 p . T 1 e ya t r R c l - h s m i n o w u d

d n r co i a l t m 2 9

s r V i oa u e d t n c v i l o s u t h w o n e r t o w d t s a i nr c b e r d i v f o m t h e n s i o l u s . i F g i ng o e r t w e i dl ob s m n e t h r a l og p l i a on t s r e i dl i ob t l y m i s bi ne d c om i n a nd l i c y a i ob e t s l m n rd j w u v r e l a og t h i p n j u v c s i ol S n ut t h i w e l a r i n or / a nd ng t i a r o r as t c o u c a n e s a m y w b us i ng o e r t o w pl e x c m bs i w t h r o a P r t 1 . 0 6 - i n F eg s d

r e s g of e r c n- v 2 .1 a n d 2 . b y i n t r o d u c l o p. r F e , a m x on a l i c r nt j oi c i l ng ra e p a b l i n e c d ar s m p h o p. l o a m e b te n r d a g i n e t h i a ng t r s c o u e - p

t i s n l d o e u r ca v . N 2 1 on N e d r ai s v t c on l u d a t c e u b i m l r s np i t h e c o . T y

∑

t i d eon c h a m T he t pl s i m on u f pe y t a t e d b l i m c a n b e er d a t n g b y u s i n g t o w t e d u a c s i M Gi =S Gi =6 e nt d p l o ps a nd us t m l f i u t he : i ons t c d (R

G1

∩ R

i w t h a r ne l s t o u c nd a a t l i d on u e av p 1

f i = 2 + 6q

q nt e d i p o s l

F

. ←

G

-G 1

2

-G 3

i w t h a n l i d o t c u- na pl e m bs l i m h i w t

d a n t e d ua c n G2

∩ R

C pe t y - b. l i m T he r t bs i m l f or

) =(

G3

d e it r n c o s a v l u f o t h e e d a t c l u n o i bp s m s r V i l o n ua t c e d . T b h g on e t h e fd i x a s e b or on a o i ng v m . l i nk or F pl e , xa m b s p e o f t i y m d l u a c 2 .3 w h bs l i m of pe t y . Tl e 2 a n b s d i t I n tr l e a mo b i s 2 .4 e h T s o b i e t s l m r e a h e t a l t i o n r o t i n m o f 2 .4 a ) r o u n d h e t s a x n g i p b y t h e c n r o f e t h n tj a c e o s e t h . n k l i h e T n t a l i r o b e s m a r a nd 2 B - 3 B i n h e t o n t s i l u n i . i F g 2 .4 b T h e a t l r u s c a m p r 2 . a bT l e i n d t r s p o u o n N e d a it r c s v o n l u f p e t y a t e d by us i ng t o w e d a c pl x m bs i w t h at ed C - p e t y b l i m . ( F g 2 5 ) a cE h r m l o g e p a b i c o m o b i l e t s m . e T h a t l r u s c r e st p a m o f e t h s u s o l 2 .3 T l e a b

∑

v 1 , ωα ). ntiquaoE 1.6)( esivg

i nR F )=

q=2 S p

a nd (

=2 F

f i = 14

1

f or t he on

pe t y - a r l s i e c m h n w t o

T1 R

b. i m l s r t c a on u b e d m t he t i l on u s n i . i g F P US UP

.2 3b,t o a c w e d g u i n h F

S . T he l ru ca t s e t r s a m p f o e s t h ns t i ol u e a r

pr e -

bs pe of t i y h m l n x

P S a nd

S . i n F g s how

SP

s nk l i 3

A

a nd 3

B

. ( iF g

t o w l i s p h ce ar j n o a d t pl e d c om by s i nk l

2 A-3

F

e t r s of e s t h ←

G

-G 1

M

-G 2

Gi

c a n l s o be n r - g 3

=S

Gi

=6

a nd t u- c ne s r e t h dl e i

ons i t a r e t e s d nr p i n

A

3 0 P 2 e c a r n l h smi t w d y o

F gi . 2 . 3 2 UP a d c n o i t l eu b m

o n e h t f i vm p g a r l

S -C

N o n o e r vt c s a i n d p a r e l e c a n h s mi o f t p y e s ( b ) h t i w c o p e u l d n r i a y o t m d n i e l o bm

2P , e i t s c bo n g m a

US - C

( a ) d an

2 p . T 1 e ya t r R c l - h s m i n o w u d

F gi . 2 . 4 2SP e ta r n i o l b m d s u c

S -C

d n r co i a l t m 3 1

N o n o e r v t c s a n i d p a r e l e c i a n h s m o f p et y s ( b t i ) c o hp e u l d w y n r a m b l i

2P , e i t s c bo g n m w bm

S -C

( a ) d an

3 2 P e c a r n l h smi t w d y o

F gi . 2 . 5 2 P RPa a d c n o i t l eu b m

o n e h t f i vm p g a r l

o n N o e r vt c s a i n d p a r e l e c a h smi n o f p t y e s cs

S -C

2P ( b ) t i h w c o p eu l d n r i a y o t m c b gn d i e l o m

s

Pa b eitsl a d n

S -C

( a ) d an

2 p . T 1 e ya t r R c l - h s m i n o w u d

d n r co i a l t m 3 a

r ca u t l S p e m s

a Te l b 2 . c rt S u a l

No .

oit S nul 2P . ( g 2i F 3 a ) 2 UP . ( g 2i F 3 b ) 8 4 4 1 9 2 3

e t rp a m

1 m 2 p

1

3 p

2

4 p

3

5 p 6

q 7 k

8

1

k 9

R

1

( R

12 13 14 15 16 17 18 19 20 21 2 23 24 25 26 27 28 29

( R

a

G3

oe S f n t T a b l 2 . 1 h c r m u s

2 8 2 3 3

( v 1 v , 2 v , 3 , ω, α ω, β δ ω )

0

0

G3

0 G1

7

G2

6

7

G3

2

2 0

0 12

12

2

4 0

F

0

0

F

1

f j

p

f j

f j

f j

2

p

j =1 3

j =1 p j =1

(

v 1 , ωα )

2

( v 1 , ωα ) 2

j =1

v 1 , ωα )

0

6

p

(

2

G2

∑ ∑ ∑ ∑

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

0

F

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

0

F

T

3

6

F

N

7

G1

M

S -C

3

2

l

r

S -C

6

R F) r

S -C

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( v 1 , ωα )

S

S

32

G3

) 6

(

31

) 6

r

30

G2

G2

r

M

)

G1

r

M

G1

S

S

M

2P . ( g 2i F 4 a ) 2SP . ( g 2i F 4 b )

0 3

(

US - C

0 2

k 10

e o p h t f a r c l smi n F g . 2 3 d 4

2

6

7 6

7 2

2 14

16 c rt u a l e p m s

3 4 P 2 e c a r n l h smi t w d y o

o n e h t f i vm p g a r l a

r ca u t l S p e m s

a Te l b 2 . 3 c rt S u a l

No .

oit S nul 2P . ( g 2i F 5 a ) 10 6 6 1 13 4 2

e t rp a m 1 m 2 p 1

3 p

2

4 p

3

5 p 6

q 7 k

8

1

k 9 (

12 13 14 15 16 17 18 19 20 21 2 23 24 25 26 27 28 29

(

)

R

G2

)

R

G3

)

G1

S

G2

6

S

G3

3

( v 1 v , 2 v , 3 , ω, α ω, β δ ω )

6

G1

6

6

M

G2

6

6

G3

2

2

( v 1 , ωα )

R F) 2 F

2 12 24

l F

12 24

2

2 0

F

∑ ∑ ∑ ∑

0

0 p 1

f j

f j

f j

f j

j =1 p 2

p

j =1

p j =1

0 12

12

12

12

2

3

j =1

v 1 , ωα )

0

M

F

(

2

0

T

v 1 , ωα )

6 6

F

( 6

G3

N

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

G2

r

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

r

M

oe S f n t T a b l 2 . 1 h c r m u s

4 2

r

r

32

1 15

6

S

S -C

7

6

(

cs

7

2

M

RPa

12

G1

r

31

2P . ( g 2i F 5 b )

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( v 1 , ωα ) 6

S

30

a

G1

R

1

S -C

1 3

(

s

1 2

k 10

Pa

e o p h t f a r c l smi n F . g 2 5

2 26

26 c rt u a l e p m s

2 p . T e 1 ya t r R c l - h s m i n d w o u

2. l i r d n c a mo t y

l d n ri co a t m 3 5

1 TR

e p a t r - l yw m s i n c u o h d pe t y -

1 RT e a r o t i h s ng c v u w d f t e r d. a n I h s i l o u , e f t h l i a on e r p t d j c yo n a e i s l r h u p v : i e l t s oc

ω

p a r l i s e t m c h n d y o uw

nd i a - l

on t ns a i r o c e v b - g t s i e l oc v d p n j u e ot d j a h i c - n b u v

v =ω α

1

= v ( q1 ) .

(q ) 2 α

ω

a nd 1

=ω α

( q ,q 1 ) α

or 2

v 1

= v ( q 1 ,q ) 1

a nd 2

i s n l o t u r ad e c . O 2 v 1 T he e d c o u p l l i c d ar n y o t i n m a n d

∑

ons t i l u f

o r v cs e ta n i d p 1

f i < 2 + 6q

T1 R pe t y - l a r s c h ni m w t e - d q t de n p i s l o p e t m t he i on t c d

. d e it r n c o s a v O l u i w t h r n e l a d t o l r u e v

e c t ua -

c on. t i e s h d pr a o u lS t i n w s h a e r c

T he i c ba s i on t s l u f o e d c o u p l ar i n y t m e l o r u a s c n G 1 b l i m of - pe t y i l on ut a s h t o w de nt p i s l o p M G1 =S

t p e y - e l p a r i es c m h a n

1 TR

F

G ←

1

-G 2

i w t h de -

be i o t n d a y us g R a nd P G1

=2

, M

( R F )=( v 1 , ωα ), M s i o l S n u t i w t h n e o a n d o t w r e s d g f o nr s a e it c o v f r om t he n s i l u . F g 2 6a by r e p n c ol v r a m g l o p by a s phe r l i c . F g ( 2 6c ) or a c r l i nd y r e st l f i a o cn . h m u p g 2 6 F j l i n ab c d r y o g m p e T h b y o i n t j a l h ce r s p g m

=S

G2 F

=S

G

F

, ( R

=6

G2

=2

,N F

P ⊥ R ⊥ Pa S a nd t he l i ow ng f e t : r s a m p v 1 , ωα ), ( R G2 )=(

b l i - m of pe t y 2

=3

) =(

G1

a nd

T F =0

. ( i F g 6a ) T s 2 h ov 1 v , 2 v , 3 , ω, α ω, β δω ),

. t can e b ed i vr ut e j oi n h l r - a p a l oi nt j . ( F g 2 6b) he T t a e r s d i . n p bT l 2 4 td b y o i e s n

Pa

s

) . 1 2 b a n d i l g se o F (

Pa

c

a nd

3 6 P 2 e c a r n l h smi t w d y o

F gi . 2 6 p o et s f y

o n e h t f i vm p g a r l

e r O vt c d o s a n i p l e c a n h s mi w t d e c o p u l P R-P

l d n ri c a y o t m RPa S

( a , ) P R-P

RPa

c

S ( b ) d an

P -PR

RPa

s

S ( c )

2 p . T e 1 ya t r R c l - h s m i n d w o u

l d n ri co a t m 3 7 a

r ca u t l S p e m s

a Te l b 2 . 4 c rt S u a l

No .

oit S nul P R-P . ( g 2i F 6 a )

e t rp a m

1 p

1

2

3 p

2

7

4 p 5

q 6

1

k ( R

10 ( 1 12 13 14 15 16 17 ( 18 19 20 21 2 23 24 25 26 a

oe S f n t T a b l 2 . 1 h c r m u s

R S

)

G2

)

G1

r

G2

M

G1

M

G2

R F) S F

r l

r F

M F

N T

F

∑ ∑ ∑ F

p 1

f j

p

f j

f j

j =1

7

p j =1

2

2

S

9 2

1

1

1

1 2 (

2

v 1 , ωα )

v 1 , ωα )

(

v 1 v , 2 v , 3 , ω, α ω, β δ ω ) (

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

2 6 0 5 2 6

v 1 , ωα )

( 2 5 1 2 1 0 2

10 1

s

7 9

9

2

j =1

RPa

8

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( 2 6 6 0 0 3 4 2 2 6 6 ( ( v 1 , ωα ) 2 2 3 4 9 10 2 2 3 2 0 0 2 2

G2

r

P R-P . ( g 2i F 6 c )

2

G1

S

S

2

9

( v 1 , ωα )

G1

c

8

2

k

9

RPa

2

1 2

8

P R-P . ( g 2i F 6 b )

1

k

7

RPa S

8 m

2

e o p h t f a r c l smi n F . g 2 6

1

12

13 c rt u a l e p m s

v 1 , ωα )

3 8 P 2 e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

o u lS t i n w s h r a g d e c

T he i c ba s i on t s l u f o e d c oupl a l i c ndr y on t i m a nd i a ng rt o a nd r i ne l a t ne d a i by us g a . ( i F g 2 7a ) T s h i l on ut a w nt e d i p l o et : r sam M

( v 1 , ωα (),

R F ) =(

F gi . 2 7 p o et s f y

t p e y - e l p a r i es c m h a n

1 RT

F

G ←

1

-G 2

i w t h de -

s t c or ua a n c e b obG

e r O vt c d o s a n i p l e c a n h s mi w t d e c o p u l P RPa S -

G1

S=

G1

1

6=

P ⊥ R ⊥ aS P

b l i m - of pe t y , M

v 1 , ωα ), M

S=

F

G2

=S F

G2

=2

,( R

2= ,N

F

=3

a nd

a nd

R

s n d a t h e i gl f o w p a v 1 v , 2 v , 3 , ω, α ω, β δω ), (

()=

G1

G 2 b m l i - of pe t y

T F =0

.

l d n ri c a y o t m P ( a , ) P RPa

c

S -R

P ( b ) d an

P aRP

s

S -R

P ( c )

R

P G2

=)

2 p . T e 1 ya t r R c l - h s m i n d w o u

l d n ri co a t m 3 9 a

r ca u t l S p e m s

a Te l b 2 . 5 c rt S u a l

No .

oit S nul P RPa S . ( g 2i F 7 a ) 8 7 2 9 2 1

e t rp a m 1 m 2 p 1

3 p

2

4 p 5

q 6 k

7

1

k (

10 ( 1 12 13 14 15 16 17 ( 18 19 20 21 2 23 24 25 26 a

oe S f n t T a b l 2 . 1 h c r m u s

R R S

G1

)

G2

)

G2

r

G1

r

G2

M

G1

M

G2

R F) S F l

r F

M F

N T

F

∑ ∑ ∑ F

p 1

f j

p

f j

f j

j =1

p j =1

S -R

P

P RPa . ( g 2i F 7 c )

8

S -R

P

8 2

9

9 2

2

1

1

1

1 2

2

v 1 v , 2 v , 3 , ω, α ω, β δ ω ) (

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

v 1 , ωα )

(

v 1 , ωα )

(

v 1 , ωα )

6 2 5 0 2 6

v 1 , ωα )

2 5 1 2 1 0 1

2 1

s

7 2

2

2

j =1

c

7

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( ( ( v 1 , ωα ) 6 6 2 2 3 4 0 0 2 2 6 6 ( ( v 1 , ωα ) 2 2 3 4 9 10 2 2 3 2 0 0 9 10

G1

S

r

P RPa . ( g 2i F 7 b )

2

k

9

P

1 2

8

e o p h t f a r c l smi n F . g 2 7

2

12

13 c rt u a l e p m s

4 0 P 2 e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

s i o l S n u t t h i w e o n a n d o t w r e s d g o f a it nr c o s e v

t c a n be e d i v r

f r om t he n s i l u . F g 2 7a by r e p n c ol v r a m g l o p by a s phe r l i c . F g ( 2 7c ) or a c r l i nd y r e st l f i a o cn . h m u p g 2 7 F T r a h el o g p i t w n c d j y t j b y o i l n ce ar s w p h

ut e j oi n h l r - a p a l oi nt j . ( F g 2 7b) he T t a e r s d i . n p bT l 2 5 s ot e d n by s

aP

c

aP

d ta n h

.

t i s n l d o e u r ca v . N 2 T he l e d c o u p i n ar y o t m a n d

s i ol n ut f

∑

n d i o t e rc as v

p e t y - a lr s i c m h n w

1 RT q t nde i p s l o e t m h n t c i o d

p

f i = 2 + 6q

1

i . ons rt e d a l h N u c w v

ga c n t - i

t si e c on. h d pr a u o u lS t i n w s h a e r c

s p e o f i l t n u ry d a c v T m h l e d c ar l i n d y o t i n m c a n b e t er d a n g b y i c t o n e obi l e s m f ar t p y w h i n l c f or m one e nt p d i p l o a nd e t h l r a e p s c ha ni m s t : i on

F

G ←

1

-G

i wt h 2

oup- de c

ng t he e fd i x nd a t he a s t . o r u e T h wb i m l

∑

S p 1

=2 F

a nd (

R F ) =( R

∩ R

G1

G2

) =(

f i = 8 . T he t r f s i b i m l c a n be of pe t y

R i wt h P

pe b of s t y i m l c nd

ro

P US

UP

S h i wt

T he r i ne l a r as t c o u i n he t b l i m . ( i F g ) 2 8a r o n i g v m l k . ( F ) 2 8b e Tr h b lim G 3 B . ( i F g 2 .8 c ) a r o u n d e t h x i s a s i p a n g b y t h e c e n t r t j s oi n a c e d h . l nt i k r I ob y m s t he i l ons u . g F 2 8d T r ca m p . 6 e 2 a bT l i t n s d p r 8 s i ol S n ut f re d a c v G 2 i w t h l e d c o u p ar i n y t o m d a n r e l i t u a c t ne d a i by i ng us a . ( i F g 2 9a ) or a r e bi n d c om g l p a r m l og e p 2 .7 T l e i a t n b s d r p 9 F g

m f ul s i he t ndi - c o

v 1 , ωα ). orF isth ae,c .qE )(16 esivg

G

M G2 = S c a n be ount e d m on t he xe fd i e a s b 2

M

= S

G1

=2

a nd

t he

.

=6

G2

G1

s i n a t l e r ob y m 2

o f t h e s i l n u . F g 2 8 c ,d T s t h e a r o

l i ona t m f k o f t h e t o w l i h ce ar s p pl e t d om c by s nk i

2 B - 3 B in

r s of t he i l n u . g F pe t y - l a r c h ni s m

T1 R

F ←

G

1

t or s c a n l s o be o b l i m - of pe t y P ⊥ R ⊥ Pa G

R a nd P

1

cs

S . ( i F g 2 9b) W e l c a r t h o w i dl e s ob t m Pa

Pa

cs

G

- pe y t

. T he l ur ca t s e m p of t he ons i l u

s

pe t y - a nd hr i l ob s e m

2

l b i - m of pe s t y

P ⊥ Pa

s

S

2 p . T e 1 ya t r R c l - h s m i n d w o u

F gi . 8 2 . o i tm n f p : y e s

l d n ri co a t m 4 1

N o n o e r v t c s a n i d p a r e l a e c h i n s m t i hw d e c o u P R-P

US

( a ) , P R - UP

S ( b ), P -PR

S ( c ) d an

P R-S P

p el d il d nr c ay S ( d )

4 2 P e c a r n l h smi t w d y o

o n e h t f i vm p g a r l a

r ca u t l S p e m s

a Te l b 2 . 6 c rt S u a l

No .

oit S nul P R-P P R - UP 6 2 4 6 1 2

e t rp a m 1 m 2 p 1

3 p

2

4 p

5 q 6

k 1

7 k (

10

a

G2

6

G2

0

M

F F

0 0 7 ( 2 0 6

2

3 0

F

∑ ∑ ∑

0

0

F

p 1

f j

p

f j

f j

j =1 2

j =1 p j =1

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

6 F

v 1 , ωα )

(

0 l

M T

2

( v 1 , ωα ) 2

)

F

N

2

2 6

G2

r

1

2

R r

3 5

6

G1

S

2

2

G1

(

S . ( g 2i F 8 c ) S . ( g 2i F 8 d )

5

( v 1 v , 2 v , 3 , ω, α ω, β δ ω )

G1

M

oe S f n t T a b l 2 . 1 h c r m u s

)

0

r

26

G2

2

S

25

)

R S r

P R-P P R-S P

2 ( v 1 , ωα )

G1

R

(

1 12 13 14 15 16 17 18 19 20 21 2 23 24

. ( g 2i F 8 a ) S . ( g 2i F 8 ) b

0

k 9

US

0 2

8

e o p h t f a r c l smi n F . g 2 8

1

2

2 6

7 8

9 c rt u a l e p m s

v 1 , ωα )

2 p . T e 1 ya t r R c l - h s m i n d w o u

l d n ri co a t m 4 3 a

r ca u t l S p e m s

a Te l b 2 . 7 c rt S u a l

No .

oit S nul P R-P . ( g 2i F 9 a ) 7 2 6 8 2 1

e t rp a m 1 m 2 p 1

3 p

2

4 p 5

q 6 k

7

1

k (

10

R

)

G2

6

M

G1

2

G2

6

R F) F

r

l F

M

7 9 2 1

∑ ∑ ∑

2

( v 1 , ωα )

6 2

( v 1 , ωα ) 2 6 12

6 ( 2 6 12 2 0

p

f j

f j

f j

2

j =1 p j =1

0 2

1

j =1

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

0 p

v 1 , ωα )

(

0

F

S

8

0

F

T

cs

2

2 F

N

RPa

6

G2

M

r

P R-P . ( g 2i F 9 b )

2

6

S

S

( v 1 v , 2 v , 3 , ω, α ω, β δ ω )

G1

r

oe S f n t T a b l 2 . 1 h c r m u s

G2

0

r

26

)

G1

S

(

G1

2

S

25

a

R

(

1 12 13 14 15 16 17 18 19 20 21 2 23 24

s

1 2

k 9

Pa

1 2

8

e o p h t f a r c l smi n F . g 2 9

2 12

12

14

14 c rt u a l e p m s

v 1 , ωα )

4 P 2 e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

N o n o e r v t c s a n i d p a r e l a e c h i n s m t i hw d e c o u

F gi . 9 2 . o i tm n f p : y e s

p el d il d nr c ay P R-P

s

Pa

S ( a ) d an

P R-P

cs

RPa

S ( b )

o u lS t i n w s h a e d r g c

p l e i S m n o d ei r n c o s a v t s i o l n u t o f p e t y l i c n d ar y o t m c a n o l s a b e n r d a t g e b y c o n e t o b i l e m s r t p f a b y o w p l e s i m b ta c e d u b y l i t . or s T he t r f s i b m l c a n be of pe s t y pe b of t y i m l s c nd

F

1

-G

i w t h l e d c oup 2

ng i t he f i xe d a n t he e a r n d ti a gr o - m or

P SU R T he a r i ne l s t or ua c n i he t

. ( i F g ) 2 10a or n i g v ml k . ( F 2 10b) e Tr h i n b ml l i nk n t i o b e . lr y a k I m s j h c d i n he t s l o u i n . F g 0d 2 1 T he tl ru ca s p 2 . 8 a bT l e i t n s d p r 1 0 F g G c a n s o l be t n d a i o by g us a

G ←

P h i wt G

M

G2 1

= S b l i m c a n be d ount e m on e t h d i xe f s e ba

G1

S h i wt

PU

M

= S

G2

G1

6=

nd a

he t

.

2=

e s i a n l r t ob y m G 1

o f t h e i n s l u . F g c ,d 2 1 0 i T s h e t r o a

l i ona t m f

3 A . () i F r g 2 1 0 c o u n d t a h e x s p b y

e of t r h i p - w s s l t i p e n dk b y c o m

2 A-3 A

2

r e st m f o h s t i l n u s i ol S n ut f re d a c v i w t h e d upl c o a i r n y ot m i ng a v h l r e n a

1R T

pe t y - l a r c h ni s m F d i a ng rt o s c u

G 1

l b i - m of pe s t y

P ⊥ aP

s

S . ( i F g ) 2 1 a or

←

G 1

-

2 p . T e 1 ya t r R c l - h s m i n d w o u

P ⊥ R ⊥ aP t e s np r . 2 1 a i F g o l u h f m

F gi . 1 2 0 . o i tm n f p : y e s

l d n ri co a t m 4 5 cs

S a nd G

2

b l i m - of pe t y R

P . ( i F g ) . 2 1 b he T l r s ca t u pa 2 . 9 a bT l e i n d

N o n o e r v t c s a n i d p a r e l a e c h s i n m i hw t d e c o u P US - R

P ( a ) , UP

S -R

P ( b ), P S -R

P ( c ), S P

- RS

p el d il d nr c ay P ( d )

4 6 P 2 e c a r n l h smi t w d y o

F gi . 1 2 . o i tm n f p : y e s

o n e h t f i vm p g a r l

N o n o e r v t c s a n i d p a r e l a e c h s i n m i hw t d e c o u

p el d il d nr c ay P Pa

s

S -R

P ( a ) d an

P RPa

cs

S -R

P ( b )

2 p . T e 1 ya t r R c l - h s m i n d w o u

l d n ri co a t m 4 7 a

r ca u t l S p e m s

a Te l b 2 . 8 c rt S u a l

No .

oit S nul P . ( g 2i F 1 0 a ) P U- R U P S - R P . ( g 2i F 1 b 0 ) 6 4 2 6 1 2

e t rp a m 1 m 2 p 1

3 p

2

4 p 5

q 6 k

7

1

k (

10 1 12 13 14 15 16 17 18 19 20 21 2 23 24

( R

)

G2

2

G2

M M

G1

6

G2

2

R F) F

r l

r F

M

1 2

∑ ∑ ∑

2

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( v 1 , ωα )

1

f j

p

f j

f j

j =1 2

j =1 p j =1

v 1 , ωα )

(

v 1 , ωα )

0 2

( v 1 , ωα ) 2 0 6

2 0 6 3

0 p

(

7

0

0

F

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

0

F

T

5

2 F

N

2

2 0

S

5 3

6

G1

r

oe S f n t T a b l 2 . 1 h c r m u s

G2

0

r

26

)

G1

S

(

G1

6

S

25

a

R

P . ( g 2i F 0 1 c ) P . ( g 2i F 0 1 d )

0 2

k 9

P S -R S P S -R

0 2

8

e o p h t f a r c l smi n F 2 g . 1 0

1 6

7 2

2 8

9 c rt u a l e p m s

4 8 P 2 e c a r n l h smi t w d y o

o n e h t f i vm p g a r l a

r ca u t l S p e m s

a Te l b 2 . 9 c rt S u a l

No .

oit S nul P Pa . ( g 2i F 1 a ) 7 6 2 8 2 1

e t rp a m 1 m 2 p 1

3 p

2

4 p

5 q 6

k 1

7 k (

10 1 12 13 14 15 16 17 18 19 20 21 2 23 24

a

)

G1 G2

2

G2

M

F

S r r

F

1 2

2 6 12 2

0

0

0

F

12

1

f j

p

f j

f j

j =1

p j =1

0 12

2

2

j =1

(

v 1 , ωα )

2

2

p

v 1 , ωα )

0

F

∑ ∑ ∑

(

6

12 F

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

6 l

M T

2

( v 1 , ωα ) 2

)

F

N

2 9

6 2

G2

R

P

7

6

G1

(

S -R

8

2 0

M

cs

6

G1

r

oe S f n t T a b l 2 . 1 h c r m u s

G2

6

r

P RPa . ( g 2i F 1 b )

2 ( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( v 1 , ωα ) 6

S

26

)

R S

25

G1

R

(

S -R P

1

k 9

s

1 2

8

e o p h t f a r c l smi n F g . 2 1

2 14

14 c rt u a l e p m s

2 p . T 3 e 1 y a t r R c l - h s m i n u wo d

2 .3 l i r d n c a mo t y

i l d n r co a t m 4 9

1 TR

e p a t r - l yw c m n s i u o d h

1 RT pe t y par l e ni s m a e c h i t h w l e d nc oup al ndr i c y t i o m a r t i s e v o u n g fc d . b l o n a t p e ir s l o c v n d e p o n t j u s o n e t d u a c p l e s a m t e s d np r n i i s t h s i e c o n t e t m h e t i o n t d c ω α =ω α ( q ) 2 .

a nd

no n s e ,t i l h I o u b nt j i o . e l t c y v T he xv

= v ( q1 ) 1

a nd 1

i s n l o t u r ad e c . O 2 3 v 1 T h e it nr d o s a c v l u f o e d c o u p l l i c d ar n y o t i n m a n d

∑

1T R pe t y - l a r s e c h ni m w t unq t de n p i s l o p e t m t he i on t c d p 1

f i < 2 + 6q

.

T he i c ba s i ons t l u f o l e d u n c ar o p i y t m b us i ng a t i c ne m h w ( s . i F g 8 2 b a n d , 8 .3 c d a n , 8 .1 2 d a n i , 3 a 8 .1 9 .3 d a n 9 .4 1 n i r t a P 1 ) o r i w t h ( ( s e f o r p l e x a m s . i F g , - 9 4 e .6 9 3 i n a P r t 1 ) t i o n . F g 2 1 a h s e tb m i l l r s ca t u e t : r s a m p ( M G2 =S t h e n s t i l o u n i . i F g 2 .1 b a n d c , b i m l . ( i F g 2 1 b) a nd l i o w n g r a t l u s c e st : r a m p ( M G1 =S . 2 10) Tl e a b T h e t l a s o j r u i v n f bw m p . i g t n F e s d o pr l u b h m r T e h s d g of rt a i n c s e v l y o a t d i n b l ip e m x a c n o d y t h r

t pe y - e l pa r s m i e c ha n

R1 T

F

G ←

1

-G

i wt h 2

s or c a n e i b t d y G 1

b l i m - of pe t y

R . T he b l i m G )=( v 1 v , 2 v , 3 , ωα )and P

R

G2

R

G2

=4

v 1 , ωα ), M

, ( R F ) =( P

G1

=2

, M

G2

1G

R =S

R G2

=5

) =( v F

R ⊥ R

, ( R F ) =(

=S

=( )

c a n be a pl e s i m or pl e x c m i - k M

=S

G2

( s e r f o pl xa m

=4

G2

n d , 8 .2 0 c 9 4 , k i - .3 7 v 1 v , 2 v , 3 , ω, α ωβ ) dan

. r o F pl e , xa m t h s ol uP P P ⊥ ⊥ ⊥ ║R 2 R G2 ( ) = v 1 v , 2 v , 3 , 1 , ωα ), ( , N F = 2 a nd T F = 0 ( s e Tl ) a b . n 2 10 I F =2 G 2 i s of s pe t y . i g ( F 2 1 c ) e Ts h on ut l a v - f R G1 ()= v 1 , ωα ), ( R G2 =() v 1 , ωα ), M F =S F =2 , N F of pe t y

G R

G2

2

l e a x s .Tr h no 2 .1 uc t e d r o i n a c h e l - p G 2

.

a nd he t l i f ow g ωα ), P

M

M

=S

G2

G1

=S

R ⊥ P ⊥

=5

G2

G1

║

=2

R ⊥ R

v 1 v , 2 v , 3 , ω, α ωδ ), = 1 a nd T F =0 ( s e

,

5 0 P 2 e c a r n l h smi t w d y o

F gi . 2 . 1 poit en f y s

o n e h t f i vm p g a r l

e r O v t c d o s a n i p a r e l a e c h i n s m t i hw u c n o p e l d

l d n r i c a y o -m P R-P

PR

( a ), P R-P

RP

( b ) d an

P -PR

R

( c )

2 p . T 3 e 1 y a t r R c l - h s m i n u wo d

i l d n r co a t m 5 1 a

r ca u t l S p e m s

a Te l b 2 . 1 0 c rt S u a l

No .

oit S nul P R-P

e t rp a m 1 p 1

2

3 p

2

4

4 p 6 7

5

0 2

( R R

)

G2

4 0 G1

2

G2

4

R F) S F

r l

r F

M N

0 2

∑ ∑ ∑

5

( v 1 , ωα ) 2 0 4

( 0 5 2

2 p 1

f j

p

f j

f j

j =1 2

j =1 p j =1

v 1 , ωα )

2

1

0

F

v 1 v , 2 v , 3 , ω, α ωδ )

0

F

T

(

2 F

v 1 , ωα )

(

5

G2

M (

2

( v 1 , ωα ) ( v 1 v , 2 v , 3 , ωα ) 2

0

M

a

G2

G1

r

26

)

G1

S

25

G1

2

S r

1 2

0 2

k (

7

2 1

k

8

. ( g 2i F 1 b ) . ( g 2i F 1 c )

7

1 k

RP R

2

q

10 1 12 13 14 15 16 17 18 19 20 21 2 23 24

P R-P P R-P

6

5

9

. ( g 2i F 1 a )

PR

6 m

2

e o p h t f a r l c mi n s F . g 2 1

0 2

2 4

5 6

7

oe S f n t T a b l 2 . 1 h c r m u s

c rt u a l e p m s

t i s n l d o e u r ca v . N 2 3 T he no i r d s a t c v l u f l e d u n c o p ar i y o n t m a d

∑

c a n be i t ne d a ob by us i ng a

pe t y - l a r s c m h ni w

1 RT q e nt p d i s l o e t m h i on t c d p 1

f i = 2 + 6q

. T he s i c ba ns t i ol u G

F 1

b l i m - of pe t y

←

G 1

-G P

2

a c t e d u by l r i ne a ot r s m R . eT h b l i m G 2

c a n be a

5 2 P e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

pl e s i m or pl e x c om a t i c ne m k i n c ha h i w t ( M . t i p l F T e g 2 1 s3 d n r bh x a m v . ( i F g 2 13a ) , l i f ow ng ur s ca t e : m p (R

R =S

G2

G2

e ( s o r f p l a e m x s . i F g ,f 1 b c e 0 a n d e , 1 0 .4

=6

G R ⊥ PS * P G2

) =(

. ( i F g 2 13b) d a n

v 1 v , 2 v , 3 , ω, α ω, β δω ), (

P M

G1

R F ) =(

=S

G1

=2

v 1 , ωα ),

R

f i n t a P r . 1) P ⊥ P ⊥ PS *

of pe s t y 2

. ( i F g 2 13c ) , d a n e h t

RS *

, M G2 =S M F =S F =2

e bs ha v l i mt o n w j d u c T pr

F gi . 2 3 . 1 o i tm n f p y e s

v 1 v , 2 v , 3 , ω, α ω, β δω ) dan

()=

G2

2G

,( R , N F =0 =6

i c o n s t .a r l e d h g

N o n o e r v t c s a n i d p a r e l e c a h s m i n i hw t c o n u

p el d il c d nr ay P R-P

PS*

( a ), P R-P

S R*P

( b ) d an

P -PR

S R*

( c )

) =(

G1

a nd

v 1 , ωα ), T F =0 .

2 . 4 a l i M r yx e g m u p c h n s t w

d n r c o i a l tm 5 3

T o w e i d l b t s m a r i n e d c o t h e ar l i p c s j P ⊥ P ⊥ S* P a nd P R

nt i o f b m l G

a n d e o n n i e t h ce ar l i s p h j i n t o o f b s l i m o f p e s t y S* R .

of pe t y R ⊥ * SP 2

P

r e 2 l u g a. x 4 i m w p M s c y h t n l i r d n c a mo t y l y i a r eM m g u x a r e t d u c by e l a r i n d g rt o a s m d a n ul c o o f t . i nr s a e c o v I e t h s n , i o l u b t h ir a o p e t o r he i nd g s p c a t u j o i e l . v T s t hi on e c ha v i l s ty f h a m

s h it ar n l o w e y p c m d -

1 RT

d e ha v r i ous e d g na l o e t s c i v a r u l e q he pl s xa m t d nr i v

=q 1

ω

a nd 1

= q 2 . W e l c a t he l a p r s c m ni of α

. ot i an l m c dr w h y

I i g s l d- o e T 2 1 R

i s n l o t u r ad e c . O 2 4 v 1 T he

∑

i ons t l u f o

v c e o tr n i s a d

i w t h l c d ar n y o t i m e v h a G 2 a r e t d u c by i ng o l s m a c . 2 1 4 p l t i s e Tg d a o m n r w F h x P R , a nd e t h g i n l f ow ( R G2 )=( v 1 , ωα a nd T F = 0 ( s e Tl a b ) . 2 1 h t i e c on r d f h a t c u pr t he G 1 b l i m s l pa r e t o he i a on rt s x of e t h a u c t he G 2 b . l i m T h e s p l e x a m i n . i F g 4 b 2 .1 n d a c e h a v a P R ⊥ P ⊥ ║R l r s ca t u e t : r s a m p ( M G2 =S G2 =2 , ( T h e p l a m e x n i . i g F 2 .1 4 d h a s a pe t y R R ⊥ ( R G1 )=( v 1 , ωα

( v 1 , ωα ), M a nd xe s l p r b h v i mt o w f j

F

l a r e p s e c ha ni m

I sl d o e i g T 21 -R p

f i < 6 q +2

1

. T he c s i ba on t l u F

G ←

1

-

be d a i o t n y us g ahs s t l r u c a : e st r a m p ( ) , M G1 =S

, M

=2

G1

=S

G2

=2

G2

G 1

R

v 1 , ωα ), M

, ( R F ) =(

F

pe b of t l y i m

v 1 , ωα ), , N F =4

()=

G1

=S

=2 F

ac t i s m j o n f t e d ol u v r j nt i of G

. i g ( F 2 14b) nd a R 1 G ) =( v 1 , ωα ), M

R F ) =(

P ⊥ P ⊥ ) , ( R G2 ) =( ║

=S

R . ( i F g ) , 2 1 4 c d a n e t h g i no l f w v 1 v , 2 v , 3 , ωα ), ( R , N F = 2 a nd F =S F =2 G 1 b i m l of pe t y i n g l f o w l r s c a t u r e st : p a m , ω, α ωβ ), M G1 =S P

F

=2

, N F

║

R , a nd e h t v 1 v , 2v , =1

a nd

3

R

T F =0

R

v 1 , ωα ), T F = 0 ( s e bT l e a 1 ) . 2 P R a nd a

()=

G2

G1

=2

, M

( s e e a bT l ) . 2 1 T h e t l a s o l u t e v r = N PQ H

b l i m of pe s t y 1

.

G2

M

=S

G1

G =S

G2

2

=5

G1

=4

b i m l of , ( R F )=

,

5 4 P 2 e c a r n l h smi t w d y o

F gi . 1 2 4 . c a l o i t n m f p et : y s (d )

o n e h t f i vm p g a r l

e r O vt c d o s a n i m i a x l y r e u g p a r e l c i a n h m P R-

P ( a ), P RP -

P - ( b ) d an

P R-

s m t i hw l c d n r - i y P ( c ) d an

P R-

RP

2 . 4 a l i M r yx e g m u p c h n s t w

d n r c o i a l tm 5 a

r ca u t l S p e m s

a Te l b 2 . 1 c rt S u a l

No .

oit S nul P R. ( g 2i F 1 4 a )

e t rp a m

1 p

1

2

3 p

2

2

4 p 5

8 9

(

10

(

1 12 13 14 15 16 17 18 19 20 21 2 23 24

R

)

G2

)

G2

0

G2

2

F l F

M N

∑ ∑ ∑

( 2

1

f j

p

f j

f j

j =1 2

j =1 p j =1

0 4

5 2

4 p

2 0

2

0

F

5

v 1 , ωα )

2

F

T

1

0

0

2

4

2

2

2

5

4

6

7

oe S f n t T a b l 2 . 1 h c r m u s

c rt u a l e p m s

t i s n l d o e u r ca v . N 2 4 T he e h ao t v i n l m c d r w y

no e d r ai s t c v

s i ol n ut f

∑

e l pa r s i c m h n

1 R e l 2- I i T s d og 1

p

v 1 , ωα )

0 2

2

2 F

(

0

4 ( v 1 , ωα ) 2 0 2

v 1 v , 2 v , 3 , ω, α ωβ )

5

0 2

v 1 , ωα )

( 2

G2 G1

(

v 1 , ωα )

( 2

2

v 1 v , 2 v , 3 , ωα )

(

0

S r

0

( v 1 , ωα ) ( v 1 , ωα ) 4

R F)

1 2

2

2

M

7

0

2

M

r

1

0

r

(

1

G1

r

26

G1

7 5

6

G1

S S

25

a

R

P

2 2

k

RP

2

4 0 2

P R. ( g 2i F 1 4 d )

2

1

k

P

6

2

k

7

P RP . ( g 2i F 1 4 b ) P R. ( g 2i F 1 4 c ) 4

q 6

P

4 m

2

e o p h t f a r l c mi n s F . g 2 1 4

f i = 6 q +2

m r oe d f i v T c a n b h . y

5 6 P 2 e c a r n l h smi t w d y o

F gi . 1 2 5 . d n ri l c a o i tm n o f t p y e s P R-

o n e h t f i vm p g a r l

o n N o e r v t c s a n i d a i x l m y r e ag l u p a r e l e c m

a h s i n m t i hw c - y P R-

R*P

( d )

*CPR

( a ) , P R P S- *

P ( b ), P R S * -

P ( c ) d an

2 . 4 a l i M r yx e g m u p c h n s t w

d n r c o i a l tm 5 7 a

r ca u t l S p e m s

a Te l b 2 . 1 c rt S u a l

No .

oit S nul P R. ( g 2i F 1 5 a )

e t rp a m

1 p

1

2

3 p

2

4

4 p 5

q 6

1

k ( R

10 ( 1 12 13 14 15 16 17 ( 18 19 20 21 2 23 24 25 26 a

R S

)

G2

)

G1

r

G2

M

G1

M

G2

R F) S F

r l

r F

M F

N

F

T

8

2 1

1

6 6

8 1

2

2

0

0 2

2

∑ ∑ ∑ F

p 1

f j

p

f j

f j

j =1 2

j =1 p j =1

v 1 , ωα )

(

v 1 v , 2 v , 3 , ω, α ω, β δ ω )

(

v 1 , ωα )

2 6 0 0 2 6

v 1 , ωα )

2 0 6 2 0 0 2

6

2

6

8

8

8

oe S f n t T a b l 2 . 1 h c r m u s

t h e d a i nr c o s v l u b y g i d l e o t m a r e bi n d c om . F g 2 15 , t w i n . b a d c T h e o n t i l u s i n . i F g 2 .1 5 a h a s a pe t y R l c ar i n d s . p T h e u r f o e l i d n a t r o s b i l m l e r u n i s a v t j . o i h e T n s o t i l u n . i F g b 2 .1 5 a n d c

v 1 , ωα )

v 1 v , 2 v , 3 , ω, α ω, β δ ω ) (

(

( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( 6 6 2 0 0 0 0 2 6 6 2 ( ( v 1 , ωα ) 2 2 0 0 6 6 2 2 0 0 0 0 2 6

G2

r

P

2

G1

S

R*P

2

6

( v 1 , ωα )

G1

P R. ( g 2i F 1 5 d )

6

2

k

9

P

4

0 2

8

P RPS * . ( g 2i F 1 5 b ) P RS. ( g 2i F 1 5 c )

2

k

7

*CPR-

6 m

2

e o p h t f a r l c mi n s F . g 2 1 5

c rt u a l e p m s

e s . i our F l d b e t s m a nd o e i . F g 2 15 G ⊥ *C

⊥ *C

⊥ P

R s i r e .T pa h

1

*C

b l i m - of pe t y P ⊥ *C

R a nd a G

- h c t ij y a w o l e r n u s v f m e t s i a r d uc t e i o n hi s t e av h a G 2

b l i m - of pe t y

2

- b l i m of

5 8 P 2 e c a r n l h smi t w d y o

o n e h t f i vm p g a r l

P a nd R

G 1

- b i m l of pe s t y P

R ⊥ S* P

. i g ( F 2 15b) a nd P

2 .1 5 c ) T o w e i d l b t s m a r i n e d c o h t s p bs . l i m T he i on ut s l . F g 2 15d ha s pe t y e ha v l p r x s a nd i s l a e p r t o t he b. l i s tm j h of e n r u v

G R

R ⊥ *R

⊥ P ⊥

⊥

P ⊥ Q P= N H

║

1

b l i m - of pe t y

R a nd P

R . T he t l a s t e ol r u v nt j s oi of e t h t o w bs i m l . he T t d ua c ol r v j i n f t he

. i g(F

S* R S*

f o es t h b i m l of

2- G G

x 0 y 0 e . pl a n - ne O i dl e i ob t l y m i s bi ne d m c o i n t he i r t d h T he l r ca t u s e m p of t he i ons u l . i F g

2 .1 T l e a b

R

c a l i e r tj o n

5 2 .1 e a r d n t s p i

2

b i ml

r 3 e Ot h

1RT

T he ons t i a c p l h e r qu g t a body n a l a n i s x ha t i s a r l c u e nd p o t e h on t r i dc f t e t l . a 19 2, ; 6 K ong d a l s i G e 2 ) . 0 T h y m e nt d p l i a o s r ( i n g r t m p l f oa h e n w t h e c o n t r i e d f o h e t i a o n t s l r i s . on a rt x

m s i e n a p hl c r - t y

p e t y - l a r s c m h n i pt e d h i s e cr a p t

1 RT

r e a f ul s i n i ne d a t g r o i d un e h i a on t s l r ye w ( R dg e a k i b lp o s n i ) a nd o e nt p d i o a r (

1T

) of t he ov- m

1R i s a r c ul ndi pe t o t he

Ts e h

pe t y - l a r s c h ni m v ob t y

1 RT

M

i t y v e n b w t h i n g o v m a d e f i x s oa r t m p l a t l e ir o n c y v r e t o a c s p e n i v g b y q . E ) ( 1 n s t i o l u e d np r i s t h i e c o n , h t i e c o n r d t l he a o p r s i d

. T he s a i b of e t h op-

S F 2= si (

R F ) =(

F

2=

a nd e c - o v 1 , ωδ ). nI the

f o e t h on s l i r a t s c onx 0 e t o l a hr i s p n x d -

z 0 a xi s . -

e s od l a t b ci m n r k h , I

v 1   = ω δ 

he r w v

t poi n i l t oc y f e h ng v , mp ar t o t j s oi n a d i s . on a t x e r h d

H f o he t i ng ov m , pl f ar t

=x 1

[ J]

t i se c r a h

ω J

2 ×2

δ

G

=2

i s e t h nd p i a t l o r e v q 2 a r e t h i l s oc v f e t d ua H i s t s u- i

pe t y - l pa r e s i e c m ha n r e a f o pe s t y v 1 , ωδ ). To ibtnao a no atdreun

T1R S=

=δ

i s t he i J a n c ob t r x. m T he i s c r t a h poi n

T he i c ba s bs i m l of G

( 3 .1 )

i s t he nd i p a l r o s t c y e v f t h c q 1 a nd

M i wt h e l pa r , s c m h ni a b s i c l m of pe t y a t e d t h i w a t t e a s l one pl e s i m or pl e x c om b i m l i n w h bi n g c om e l s oc i v t a c hi n g m s , a c p l o t i e h i s c r t a n po r c e nt poi a nd s i e d ua t on t he i s a x of e t h j oi r t m p l f oa b l i m w t h t a l ue ; i L 0 2 j o . M h w r s c P

q 1  q   2 

2 ×2

a nd (

R F ) =(

P ⊥ R pe t y -

T1R P ⊥ R . ( i F g a P r t 6 - 1) i s - oc a

i ch 3

ωδ in eth isba fo tsi onalitper e.pacs nI

v 1 a nd

≤ M

Gi

=S

Gi

M

H i de s nc o t h i w t he o l nt i e c g o t h n v m Gi

=S

Gi

= 3 a r o ug ( z B e t l . a 2 0 ; t h b C e l . a ) . 0 6b,2 7

G. Gogu, Structural Synthesis of Parallel Robots, Solid Mechanics and Its Applications 183, DOI 10.1007/978-94-007-2675-8_3, © Springer Science+Business Media B.V. 2012

≤ 6

6 0 3 T e r O h t p 1 R - ya c l n s m i

3 .1

1 TR

e p a t r - l yw m s i n u o c h d

n I t he s t hi n, t si e c o t he t o w a t l i on pe r s i e l t oc v d e p n e l t s : oc i v

pe t y - l pa r e s c h ni m w t h

1 RT

e d n tr s p i

l e d c oup s t m i on on bot h t e d ua c t j i n o

v 1

= v ( q 1 ,q ) 1

ω

a nd 2

=ω δ

( q ,q 1 ) δ

. 2

i s n l o t u r ad e c . O 3 1 v T he e o r vn cs ta i d e d c oupl s ot i n m nd a

∑

p 1

f i < 2 + 6q

s i ol S n ut of pe t y a nd r i e l a or i a ng t s c or u a n e b t n r d. a g t o w e d ua c bs l i m t h w pe t y < hi w c 3

i l ons ut of

pe t y - l e pa r s e c m ha ni i w t h

T1 R t de n p i o ps l e t m t he i on t c d q

. F ←

G

-G 1

-G 2

i w t h a s r i v o u s r e d g f o t a ei r n c o s v 3

e h T i c ba s l on ut s e M

Gi

=S

a nd a n l i o t d e ua n c b l i m of

=3

Gi

P ⊥ R . e rd D i v s l on ut c a be i t n d a ob y ng us i a t c S Gi N F >0 i ng c u o e t o e f i v dl s e d r o uc t i n c h e a y-

i n g o e tc l j r u v b y i t l y s c e d r i o u nt t he g l ru s ca t e : m p M F

G3

) =(

=S

=2 , ( R v 1 , ωδ ), M F =S G3

()=

G1 F

=2

v 1 v , 2 v , 3 , ωδ ), ( , N F = 5 a nd T F =0

R

.

G2

)=

3 p . T 1 e ya t r R c l - h s m i n o w u d

F gi . 3 1 b i l m a d n c o p eu l t si m n o f p : et s y =5 N

ons 61

e r O vt c d o s a n i

e p c a r n l h smi w t d o u

1T R

2P ( b ) d an

P RS * -P

RP -

hti w

N =4

R -P (c )

t i hw

=6 N

( a ) , P * CR - P

R -P

t i hw

6 2 3 T e r O h t p 1 R - ya c l n s m i

F gi . 3 2 b i l m d a n c o p eu l d i t s m n o f p et : y P R ti h w

e r O vt c d o s a n i

e p c a r n l h smi w t d o u

1T R

P * CS - P =2 N

( b ) d an

P * CS - P

RS * -P

hti w

R -P ti h w

=1 N

(c )

N =3

( a , ) P * C S- P

* CR -

3 p . T 1 e ya t r R c l - h s m i n o w u d

F gi . 3 b i l m a d n c o p eu l t si m n o f p : et s y =5 N

ons 63

e r O vt c d o s a n i

e p c a r n l h smi w t d o u

1T R

2 RP ( b ) d an

RP

S * -RP

PR-

hti w

N =4

R-P (c )

t i hw

=6 N

( a ) , * CP

R- P

R-P

t i hw

6 4 3 T e r O h t p 1 R - ya c l n s m i

F gi . 3 4 b i l m a d n c o p eu l t si m n o f p e : y s P R ti h w

e r O vt c d o s a n i

e p c a r n l h smi w t d o u

1T R

* CP =2 N

( b ) d an

* CP

S * -RP

S * -PR

S * -RP ti h w

ti h w

R-P =1 N

( c )

N =3

( a ) , * CP

S * - CP

R-

3 p . T 1 e ya t r R c l - h s m i n o w u d

ons 65 a

r ca u t l S p e m s

a Te l b 3 . 1 c rt S u a l

No .

oit S nul 2P . ( g 3i F 1 a ) 2 RP . ( g 3i F a ) 7 3 3 2 8 2 3

e t rp a m

1 m 2 p

1

3 p

2

4 p

3

5 p 6

q 7 k

8

1

k 9 1

(

12

R

(

13 14 15 16 17 18 19 20 21 2

R

G3

)

R-P

R -P R- P

R-P

7

3

3

3 2

8 2

2

3

3 0 3

( v 1 v , 2 , ωδ )

3

v 1 v , 2 v , 3 , ωδ ) ( (

v 1 v , 2 , ωδ )

(

3 0

0

r

G2

0

0

0

r

G3

0

0

G1

3

4

5

M

G2

3

3

3

G3

2

2

2

0

∑ ∑ ∑ ∑

7 2

6

5

0

F

p 1

f j

f j

f j

f j

j =1 p 2

p

j =1 3

j =1 p j =1

8

2

F

T

2 0

6 F

v 1 , ωδ )

2

v 1 , ωδ )

2 0

M N

(

2

F

(

0

M

r

v 1 , ωδ )

2

0

F

( 3

2

l

v 1 v , 2 , ωδ )

5

G1

r

v 1 v , 2 , ωα , ωβ , δω ) (

v 1 , ωδ ) (

2

( v 1 , ωδ )

-PR

2 8

4

R F)

- RP

7 3

( v 1 , ωδ )

G3

P RS * -P . ( g 3i F 1 c ) R P S * -RP . ( g 3i F c )

3

( v 1 v , 2 , ωδ )

G2

S

oe S f n t T a b l 2 . 1 h c r m u s

)

S

(

32

G2

G1

M

31

)

S r

30

G1

3

S

23 24 25 26 27 28 29

a

R

P * CR - P . ( g 3i F 1 b ) * CP . ( g 3i F b )

0 3

(

R -P

0 2

k 10

e o p h t f a r c l smi n F g . 3 1 d

4

0

0

3

4

5

3

3

3

2

2

2

8

9

10 c rt u a l e p m s

6 3 T e r O h t p 1 R - ya c l n s m i a

r ca u t l S p e m s

a Te l b 3 . 2 c rt S u a l

No .

oit S nul P * CS - P . ( g 3i F 2 a ) * CP . ( g 3i F 4 a ) 7 3 3 2 8 2 3

e t rp a m

1 m 2 p

1

3 p

2

4 p

3

5 p 6

q 7 k

8

1

k 10 ( R

1 ( R

12 (

23 24 25 26 27 28 29

R

a

)

S * - CP

P * CS - P . ( g 3i F 2 c ) * CP . ( g 3i F 4 c )

R-P

7 3 3

3 2

2

3

3

0

0

G3

)

3

v 1 v , 2 v , 3 , ωδ )

(

( v 1 , ωδ )

v 1 , ωδ ) (

6 4 0

0

r

G2

0

0

0

r

G3

0

0 6

6

M

G2

3

4

5

M

G3

2

2

2

( v 1 , ωδ )

(

2 0

0

9 F

∑ ∑ ∑ ∑

10

2 F

p 1

f j

f j

f j

f j

j =1 p 2

p

j =1 3

j =1 p j =1

2 2

0

F

1

2 3

F

v 1 , ωδ )

2 0

l

1 0

0

6

6

6

3

4

5

2

2

2

1

v 1 , ωδ )

0

6

2

(

2

0

F

v 1 , ωδ )

5 2

R F)

( 6

2

G3

v 1 v , 2 , ωα , ωβ , δω )

(

G1

T

* S- P R

8 2

M

N

S * -RP

2 8

G1

r

* R S- P

7

3

( v 1 v , 2 , ωδ )

3

M

oe S f n t T a b l 2 . 1 h c r m u s

G2

G2

S

r

32

)

G1

S

r

31

R-P

* CR - P

3 ( v 1 v , 2 v , 3 , ω, α ω, β δ ω ) ( v 1 v , 2 v , 3 , ω, α ω, β δω ) ( v 1 v , 2 v , 3 , ω, α ω, β δω )

G1

6

S

S

30

S * -RP

P * CS - P . ( g 3i F 2 b ) * CP . ( g 3i F 4 b )

3

k

13 14 15 16 17 18 19 20 21 2 (

R -P

0 2

9

e o p h t f a r c l smi n F g . 3 2 d 4

12

13 c rt u a l e p m s

3 p . T 1 e ya t r R c l - h s m i n o w u d

T he rd i v ons ut l . g F 3 1c d a n . r e ob ns t i ol u . F g 3 1a d by r e p c c a l t j . i n o s T , hu t o w na l i r e d t s l i ob m r e a c a l t j i n o *. S ot B h ns i o ut l e ha v t he ng l i f ow s t r M G 1 S=

ons 67

e d a i t n r om f h c ba s ut e ol j i v n by a s ph r duc e o t r i n e t h r i p - s , M

5=

G1

( v 1 v , 2 , ωδ (), s r V i oa u e d i v r o n s t i l u c a n b e e d a i o b t n b y b i c n o m a nd l i o rt e d obi t s l m n bs l i m l r s ca t u e m p f o e t h ns i ol u s . i F g 3 1 d 3 .2 1 a b l n e s R

S=

G2 G3

, M

3=

G2

v 1 , ωδ ),(

()=

S=

G3

,( R

2=

G3

) =(

G1

v 1 , ωδ ), M

R F ) =(

r u c a t l : r e st a m p v 1 v , 2 , ωα , ωβ , δω ), (

=S F

,N

=2 F

a nd

=4 F

T F =0

R

G2

)=

.

i ng l ona t i r s G 1

a nd G

( i F g s . 3 2 a n d 3 .4 ) T h e 3 . 4 - a r e t s d n p i a -T 2

o u lS t i n w s h r a g c

s i ol S n ut f pe y i n g r t o a o r u a s t c n c a i w s e l k b e t er d a n g b y u s i b s l i m t h i w i a n g r t o o r s t u a c t e s d np r i n s . i F g 7 . or F pl e , xa m t h b l i m of pe t y i n g s ta c o r u t e d s u a i o n h e t e f d i x e . b a s T h e o l e v r b s e l i ah v m . T p r x t f o t he on t s i l u ( R F ) =( st : e r ( R G3 e d s a it nr l o u i n h w c s t i on by c r g du e f v l i t s m ( e i d l o b i l t m y i s e d r o u c n t i i n e a c h l i c n d ar y j i o n t j * . a S o i l s p h ce r T he s . i F g 3 5 a n o eb y r c p l j u t v i n l o ut he .T s b y d m a r g l r s c a t u : r e st a m p ( v 1 v, N F =5 T he d i v r n t s ol u . i F g 3 5c s d i bt ne a o f r m s . i F g 3 5a by i ng ra c e pl o e l r ut v j i n o by a s ph i ng t o w a r l e d i b s . m T h n t l o u i n g a t l r u s c e st : r a m p ( v 1 v, S F =2

F

G ←

-G 1

-G 2

3

i w t h oa u s r v e g d f o t a i r n c s v d g n pl e s i m or pl e x c om c a - 1 4 a n d a P r t 7 .1 6 - 1 . i n . F g 3 5a s u e a n l i d o t e a c u n

b a sc i u o t l n P ⊥ R a nd t o w pl e m i s r a n b of pe t y

R

i wt h a r o -

║R ║R

e ut j t s oi n f o e t h e t hr e sn t , d l p i o a

q=2

i n . i g F 3 .5 a l f s i u t h e s : i n d o t c R G1 ∩ R G2 ∩ R G3 )=( v 1 , ωδ ). tI has the inglfow rsatluc e-parm M G1 = S G1 = 3 , M G2 S = G2 = 3 , M G3 = S )=( v 1 , ωδ ), ( R F )=( v 1 , ωδ ), M F =S F =2 , N 6 > N F > 0 c a n be d i v r om f t he s i c ba s ol u-

S =2

G3 F

=6

, ( R 1 G ) =( a nd T F =0 R

G2

F

=2

a nd

v 1 v , 2 , ωδ ),

=( )

. t O he r c on- v

e s . i F g 3 5 a nd 6) O t n * C a nd t o w i n c h e a i n . F g 3 5b i s t ne d a o r m f h c b s i t l on u

d ie r v t l s n u o

ni a l i t j c dr b n * - C o m l i f o w t n g . h3 e 5 b a s F

R 2 v , 3 , ωδ ), ( a nd T F =0 .

G2

M G1 = S G1 = 4 , M G2 = S G2 = 3 v 1 v , 2 , ωδ ), ( R G3 ()= v 1 , ωδ ), (

()=

, M R F ) =(

G3

= S G3 = 2 , ( R v 1 , ωδ ), M F =S2

G1

)=

t he ba s i c on l u

2

a l i ce r t j n o * S b i n - c o m , ωα , ωβ , δω ), (

,N F

=4

a nd

T F =0

M R

G2

.

) =(

G1

=S

G1

=5

v 1 v , 2 , ωδ ), (

. i F g 3 5c ha s e t l - f ow , M G2 = S G2 = 3 , M G3 = S R G3 =() v 1 , ωδ ), ( R F )=(

G3

= 2 , ( R G1 ) = v 1 , ωδ ), M F =

,

6 8 3 T e r O h t p 1 R - ya c l n s m i

ai ons t re d V l h u c w v by bi n g c om l i ona t s r a nd l i a on rt l e i d obi m a nd G e s a m l u r s c a t e t r s a m p a s r e i t h t sa r e u n p c o i w . d 3 2 ) s 1 a n e bT l ( i F g

F gi . 3 5 b i l m a d n c o p eu l d o i t n s m o f p : et s y =5 N

4 > N F

0>

c a n i be t d o

t s i e l i n t he bs l i m 2

. ( i F g 3 .6 ) e T h n s u t i l o d n t e s p r i n s . i g F 3 .5 a t h r i ne l a or uas t c i n

e r O vt c d o s a n i

e p c a r n l h smi w t d o u

1T R

2R ( b ) d an

G n d 3 .6 e a v h t h e

R RS -

R -P

ti h w

=4 N

R -P (c )

hti w

=6 N

( a ) , R R -C

R -P

ti h w

1

3 p . T 1 e ya t r R c l - h s m i n o w u d

F gi . 3 6 b i l m d a n c o p eu l d o t si m n o f p : et s y t i hw

ons 69

e r O vt c d o s a n i

1T R

e p c a r n l h smi w t d o u R S C- R

=2 N

( b ) d an

R S C- R

RS -P

ti h w

N =1

R -P (c )

i ht w

N =3

( a ),

R S C- R

R -C P

7 0 3 T e r O h t p 1 R - ya c l n s m i

t i s n l d o e u r ca v . N 3 1 2 ons i l ut w h a r i ne l o g t i a r s c o u a nd

on N e d r ai s v t c l i d o n a t e d u a t c n b i m l e a r t e s d np r i n h i s t t s e c

∑

on t c i d e h a m ps l T he t pl s i m on u f pe y t a t e d b l i m c a n b e er d a t n g b y u s i n g t o w t e d u a c s i M Gi =S Gi =6 t e nd i p ops l d a n us t m f i ul he t s t : i d on c (R

G1

∩ R

F ←

G 1

a don. i e Ty h e ha v p 1

-G

f i = 2 + 6q 2

-G 3

q i nde .

i w t h a n l i d o t c u- na pl e m bs l i m h i w t

P ⊥ R pe . t y - T h te r bs i m l r f o

nd a t e d ua c n b i m l G2

∩ R

G3

) =(

∑

v 1 , ωα ). iquaonEt )(1.6 esivg

S F

p

f i = 14

1

q=2 R F )=

a nd (

=2

f or s e t h

ns . o t i l u re d a c v o u lS t i n w s h a e r c

i a n s d .3 7 u e F o gt l p h , r x m pe t y

F gi . 3 7 a et d bu il m n c op s f :y

b of l t i e m d ua n c P ⊥ R a t e s u i r c o n hl b w p m d

o e r N v c- n t d s a i

d i ex f o n t h

1T R

p a r e l c h mi n s w t a d o i l n u a c 2P

* CS - P R

( a d n) a

2* CP

S * -PR

( b )

3 p . T 2 e 1 ya t r R c l - h s m i n d w o u

oit s n 1 7

o e r N v c- n t d s a i

F gi . 3 8 a et d bu il m n c op s f y

p a r e l c h mi n s w t a d o i l n u a c -

1T R

2R

ba s e . i g ( F 3 7a ) or n a i ng ov m l k . ( i F g 3 7b) T pe s t y

* CS - P R

he pl s i m b a r e of P ⊥ *S C

) or .3 7a ( i F g

⊥ P S* ) . 3 7b ( i F g

*C

o u lS t i n s g r a c

T he i l on ut s . i F g 3 8 e s u a n l i o t d ua n c a nd pl e t o bs i w m f y s e . fd ba i x T r bs he t l i of m n u . g 3 7 F a d e nt s l o p a nd i l f u t he l i ow ng f : i ons t d c ( R F ) =( R G1 ∩ R G2 ∩ R G3 ) =( l r s ca t u : e m p M ( v 1 v , 2 v , 3 , ω, α ω, β δω ), ( R

h t i r w u a se n g d c o m .8 r f o

G1 G3

v 1 , ωα ). eTsh ilonsut eavh the inoglfw = S G1 = 3 , M G2 = S G2 = 3 , M G3 = S ()= v 1 , ωδ ), ( R F )=( v 1 , ωδ ), M

S ,( R

=2

G3

=S F

F

=2

F

G1

a nd

2= =( ) R

,N F

a nd

e p a t r - l yw m s i n c u o h d d a n e l r i o-

l c o e du p n s i m t

nt a c i e d. b r I g ona l i t i e l t oc y v of t he e l t oc y i v a nd t he a r- o δ

: i e ts c ( q ,q 1 ) 2

.

v 1

= v ( q1 ) 1

=)

G2

=0

nt j oi * C a nd t o w

1 TR

T1 R pe t y - l a r s i c m h n w t tr i s a n g c o e h u d w f v t h e i l o n s u t e d n t r s p i n t h i s i c o n , t s e e t h l nt s r a o i ng v m pl ar t f s de n o j s t u one t d ua c j oi n a t l i on c y e l o v d p n s on b t h d a t c e u j i nt o e l v ω δ =ω

i nde p -

2 q=

. ne O i dl ob t y m i s b ne d c om i h e a c r l i nd y

T F =0 n t j * . a S o i l h ce r s p

3 .2 i s n mo t

P ⊥ R

t e d b l i m of pe t y R *S C

a nd

7 2 3 T e r O h t p 1 R - ya c l n s m i

i s n l o t u r ad e c . O 3 2 v 1 T h e d a it nr c o s v l u o f e d c oupl s ot i n m nd a

∑

p 1

1 RT pe t y - l a r s e c h ni m w t de t de n p i o ps l e t m t he i on t c d q

f i < 2 + 6q

. ei r n c d o s a v t O l u i w t h r n e l a d t i a n r o

g c t ua -

c on. t i e s h d pr a o u lS t i n w s h a e r c

T he i c ba s ons t i l u f o R1 T t pe y - l pa r e m i s c ha n F e a i obt n r s c l u d m p a us i ng y d b P ⊥ R a nd t he G 2 b l ofi m - pe s t y P ⊥ R ║R .(iFg39a)or pe t y 3 .1 0 a ) h e T o l u t v r j s i n o f e t h o w b s l i m e h a v p l a r e s . x e T h r i n la r a t c o u f he G 2 b l i m - c a n e ou t d h i xe fd a s b . g ( F 3 9 ) i a ng ov l k m . ( F 3 10 ) Ts e h ut n e nt d p i o l t h e i n gl f o w a l r s t u c e t : r s a m p M G1 =S G1 =2 ( R G1 )=( v 1 , ωδ ), ( R G2 =() v 1 v , 2 , ωδ ), ( R F )=( v 1 , ωδ ), M F =S T F =0 . t O he r d i n o s a e r c v t i l on u i w t h t o w a nd one de t s a i nr c b e d v f o m h a s i n u t l i r yng oduc t e w e i d l i o b e t s l m e ( s s . i g F 3 .9 a n d ) . 3 1 0 n e O l e i d i ob t l m y a n c e b r o- nt i d u c e b y n g r a c ie p l o u t v j i n o b y a l i c n d r y l c a t j oi n * C s . ( i F g 3 9b a n d ) , 3 .1 0 b d a n t o w e i d l o b i l e t s m b y c i p l n g r a e o n e t e ol r u v t j oi n by a a l i s p h ce r n t j o i * S s . ( i F g 9 c 3 . a n d . 1 0 c ) 3 T h e o l u s n s t i o i n s . i g F 3 .9 b d a n 3 .1 0 b e a v h e t l i f o w n g r a t u s c e : r a m p M G1 S = G1 ( R G1 )=( v 1 , ωδ ), ( R G2 ()= v 1 v , 2 v , 3 , ωδ ), ( R F )=( v 1 , ωδ ), M T F = 0 , a n d o s e t h i n s . i g F 3 .6 c d a n 3 .7 c e h a v M G1 S = G1 v 1 v , 2 , ωα , ωβ , δω ), ( R F )=( v 1 , ωδ ), ( R G1 )=( v 1 , ωδ ), ( R G2 )= ( a nd

G ←

-G 1

i w t h de 2

G

b of l i m 1

R ⊥ P ⊥

║

R . ( iF g

or n nd a , M

G2

, N

=2 F

=S

a nd

=3 F

,

=3

G2

r e s g of e r c on- v

, M

=2

=S

G2

G2

=4

, N F =2 , = 2 , M G2 = S G2 = 5 , M F =S F =2 , N F =1 F

=S

F

=3

.

T F =0

T he l r ca t u s m p f o he t i ons u l s . i F g

3 .9 a n d 0 3 .1 r e a s - y

Tl e 3. a b t d i n z m o u lS t i n w s h a e d r g c

T he i c ba s ons t i l u f o e d c oupl t i ns m a y b r d g t o i m us i ng a e o pnt d l i s ha v T u et : r sam ( R F ) =(

t pe y - l pa r e m i s c ha n

R1 T

F

G ←

-G 1

2

i w t h de -

or s c a n be d i t y G 1

P ⊥ R a nd t he

b l i m of - pe t y

G 2

b l i m of - pe t y R

s a t l i r u p o w -n g c h e f d M G1 = S v 1 , ωδ ), M F

G1

=S

,

=2 F

=2

M

,N

=S

F

G2

=3

G2

a nd

=3

, ( R G1 ) =( T F =0 .

v 1 , ωδ ), (

. ( i F g 31 a )

║R ║R

R

G2

) =(

v 1 v , 2 , ωδ ),

,

3 p . T 2 e 1 ya t r R c l - h s m i n d w o u

F gi . 3 . 9 p : et s y

oit s n 3 7

e r O vt c d o s a n i P R-P

R

i ht w

=3 N

1T R p a r e l a e c h i n s m t i hw d e c o p u l o s i t n m o f ( a ), P R-P * CR

ti h w

=2 N

( b ) d an

P R-P

RS *

ti h w

=1 N

(c )

7 4 3 T e r O h t p 1 R - ya c l n s m i

F gi . 1 3 0 p : et s y

e r O vt c d o s a n i P R- P

R i ht w

=3 N

1T R p a r el c h si n m wt d e c op ul it n s m o f ( a ) , P R -* C P

R ti h w

=2 N

( b ) d an

P R- P

S * ti h w

=1 N

(c )

3 p . T 2 e 1 ya t r R c l - h s m i n d w o u

F gi . 1 3 p : et s y

oit s n 5 7

e r O vt c d o s a n i P R-

R

i ht w

=3 N

1T R p a r el c h si n m wt d e c op ul it n s m o f ( a ), P R* CR

ti h w

=2 N

( b ) d an

P R-

RS *

ti h w

=1 N

(c )

7 6 3 T e r O h t p 1 R - ya c l n s m i a

r ca u t l S p e m s

a Te l b 3 . c rt S u a l e t rp a m

No .

1

oit S nul P R-P P R- P P R5 2 3 5 1 2

m 2 p 1

3 p

2

4 p 5

q 6 k 1

7 k

8

1 12 13 14 15 16 17 ( 18 19 20 21 2 23 24

2

25 26 a

R R

)

G2

) 3

2

G2

3

5

4 (

∑ ∑ ∑

oe S f n t T a b l 2 . 1 h c r m u s

t O he r d i n o s a e r c v t i l on u i w t h t o w a nd one de t s a i nr c a b e d i ve r m o f t h i s c a s i b n t o l u s b y e i dl ob t s m e ( . i g F 3 1 ) ne O l i d ob t y m i pl ng a c o e l r ut v j n oi by a c l i ndr y j t o C o b i e t s l d c m nj g y r a u p v 3 .1 c ) T e h s s t i l o n u e a v h t h e e a m s l r s c a t u p a a n s i .3 9 r F u g t o c e l h w p

j

p

f j

f j

j =1 2

j =1 p j =1

5

2

2 2

0

v 1 , ωδ )

0 4

3

f

(

0

2

1

5

v 1 , ωδ )

2

3

p

0 2

0

F

0

2

F

v 1 v , 2 , ωα , ωβ , δω )

( 2

4

( v 1 , ωδ )

F

F

v 1 v , 2 v , 3 , ωδ )

(

v 1 , ωδ )

(

2

l

F

2

v 1 , ωδ )

(

2

M

2 0

0

S

T

1 2

2

R F)

N

5 1

0

M

RS *

. ( g 3i F 9 c ) . ( g 3i F 1 0 c ) . ( g 3i F 1 c )

3 5

G2

2

S*

2

G1

G1

RS *

5

( v 1 v , 2 , ωδ )

G2

M

P R-P P R- P P R-

3

0

r

* CR

. ( g 3i F 9 b ) R . ( g 3i F 1 0 ) b . ( g 3i F 1 0 ) b

2

0

r

* CR

5

2 ( v 1 , ωδ )

G1

G1

S

r

P R-P P R -* C P P R-

0

2

S

r

. ( g 3i F 9 a ) R . ( g 3i F 1 ) 0 a R . ( g 3i F 1 ) a R

0

k

9 ( 10 (

e o p h t f a r c l smi n F g . 3 9 - 1

1 0

0

2

2

2

3

4

5

5

6

7 c rt u a l e p m s

r e s g of e r c on- v c t i r ng odu e r o t w c a n be c d r i o u nt by r e * s . i F g ( 3 1 b) , a nd t o w a j c e r i l n s t p h .* S o g ( F r e st a m s a r h e i t ) l . e 3 a bT ( s 1 0 d

3 p . T 2 e 1 ya t r R c l - h s m i n d w o u

oit s n 7

t i s n l d o e u r ca v . N 3 2 T he n d i o t e rc as v l e d c oup ons t i m a nd

∑

p 1

s i ol n ut f f i = 2 + 6q

p e t y - a lr s i c m h n w

1 RT t de n p i ops l e t m t he i on t c d q

i . ons rt e d a l h N u c w v

ga c n t - i

t si e c on. h d pr a u T h e p l s t i m o n it r e d s a c o v l n u t o f p e t y F

e d c oupl s t i n m a c e b t n r d a g y i c n t o e h s r t m p l f oa b y t o w p l e s i m b m i l i w t h r i n e l a r n d / o a r o t o w bs i m l r m f o one t e nd i p l o p a nd e t h e l a r p t he ns t : i d o c

G ←

1

-G 2

i wt h

e- d

i xe fd a n e t h obi l m a ng t i r as t . c o u e T h

∑

e s i v ) g ( 1. 6

S 1

p

F

R F ) =(

a nd (

=2

R

G1

∩ R

G2

l i es c m h a n s l f i u v 1 , ωδ ). orF sith ,case q.E

) =(

fi = 8 .

o u lS t i n w s h a e r c

o r F p l e , x a m t h e e d a o v it r n c s - i o n s u t l h i w t . i F g 3 12 e ha v t of pe s t y o i n t j ar l c d y h e u

F gi . 3 . 1 2 o si t n f p : y e

r i ne l a or uas t c n i G P ⊥ C* S

or

1

P ⊥ R i wt h M G2 = S

b l i m - of pe t y C*

⊥ P S*

i wt h C*

o e r N v c- n t s a i d * CS

( a d an)

P R -* C P

G2

=6

= S

G1

G1

2=

a nd

t he G

. ne O l i d ob t m y i s n*S .

p a r e l a e c h i n s m t i hw d e c o p u l o -m

1T R P R-P

n t j ar o i l ce s p h w d

M

S*

(b )

2

b lim -

7 8 3 T e r O h t p 1 R - ya c l n s m i

o e r N v c- n t s a i d

F gi . 3 . 1 o si t n f p y e

p a r e l a e c h i n s m t i hw d e c o p u l o -m

1T R P R-

(a )

* CS

T he r e l a i n or uas t c i n t he G

b l i m - c a n be ount e d m on t he i xe fd e a s b 2

. 3 1 2 b ) i g n( l F o v k a m r o u lS t i n w s h a e d r g c

or F pl e , a m x t h i n c d s v o - u w t or uas i n . F g 3 1 s ha e t G 2 b l i m - of pe t y i n j ar o l p h ce t s d w y T h e n s t i o l u i n s . i F g 2 3 .1 a n d 3 .1 e h a v t h e l f o M G 1 S= G 1 2=

ne a r i l d g t o a c G R

3. i s n mo t

1 TR

M

, M

S=

G2 F

=S

G2

=2 F

, ( R

6= ,N

F

G2

a nd

2=

G1

eth

. ne O l i d ob t y m i s b ne d c om i

6=

=0

) =(

G1

a nd

v 1 , ωδ ), (

T F =0

R

G2

i w n g e t : p ra s m v 1 v , 2 v , 3 , ω, α ω, β δω ),

) =(

.

a nd a s r i vou de -

n l c o e u pd o n s i mt

s , i ol n ut h bot r a - ope . t oc y i e l v T h pl e s a m x v

i s n l o t u r ad e c . O 3 v 1 T he vc t i eo n r s a d e d c oupl s ot i n m nd a

∑

= S

G1

e p a t r - l yw c m n s i u o d h pe t y - pa r e l s e c ha ni m i w t h

T1 R r e s g o f t i nr o s a e c v n c a b e t e n r d . a g n I t h e s a t l i on s e c v nd p o j s t u e on t d ua c j oi n on t i d e h c m s pr

= S

G2

M

t.

v 1 , ωδ ), M

( R F ) =(

1

i wt h

║SC

P ⊥ R i wt h

b l i m - of pe t y

p 1

ons t i l u f q f i < 2 + 6q

.

T1 R pe t y - l a r s c h ni m w t u t de n p i o ps l e t m t he i on t c d

1

= v ( q1 ) 1

a nd

ω δ

=ω δ

(q ) 2

.

3 p . T e 1 y a t r R c l - h s m i n u wo d

e r O vt c d o s a n i

F gi . 3 4 1 p : et s y

P R-P

F gi . 3 5 1 p : et s y

P R-P

T he l e d unc o p t i s m a by r n c e b l i m of pe t y n t s i o l u s h a e t h l i o w n g f t l r u s c a : r e st a m p ( ( v 1v, 2 T F =0 or F pl e , a m x t h i l on u s . i F g 3 14b a s h t e t he G st : e r (

oit n s 9 7

p a r e l c h smi n t w c o n u p e l d i t s m o f

1T R PR

( a ) d an

P R-P

(b )

R

e r O vt c d o s a n i

p a r e l c h smi n t w c o n u p e l d i t s m o f

1T R ( a ) d an

*P C

P R-P

of

i c bas i l on ut s

(b )

*C

t pe y - e l pa r s e c m ha ni

1R T

F ←

G 1

-G

h i wt 2

i bt ne d a o y g us P ⊥ R a nd t he , ωδ ), M . ol S ns ut i h w no e d a u t c i s pr m t j o n c a

G1

=S

G , M

=2

G1

G2

2

=S

G2

=3

, ( R F ) =(

R . ( i F g 3 .1 4 a ) i T s h R G1 ) =( v 1 , ωδ ), M F =S

b l ofi m pe - t y R

G1

()=

1

-

v 1 , ωδ ), ( R G2 =) , N F = 3 a nd F =2

s o a l be d. a i o t n G

2

G ⊥

P ⊥ P ⊥

b l i m - of pe t y

v 1 , ωδ ), (

R P R

R G2

=( )

R ⊥ R e t - . u r a l h mi I s p n g c f o w v 1 v , 2 v , 3 , ω, α ωβ ),

M

1

G1

P ⊥ R a nd

b m l i - of pe t y =S

G1

=2

, M

G2

=S

G2

=5

,

8 0 3 T e r O h t p 1 R - ya c l n s m i

( R F ) =( l a e p r .3 14 h v i F g n s t o u w

v 1 , ωδ ), M F

=S F

=2

,N

a

r ca u t l S p e m s

No .

c rt S u a l

oit S nul P R-P . ( g 3i F 1 4 a ) 5 2 3 5 1 2

e t rp a m 1 m 2 p 1

3 p

2

4 p 5

q

6 k 7

1

k 9

(

a

G2

)

R

2

G2

3

S G2

F

0 0 5

0 5 2 3

∑ ∑ ∑

1 0

F

1

f j

f j

f j

j =1 p 2

j =1 p j =1

0

2

2 3

5 5

7

oe S f n t T a b l 2 . 1 h c r m u s

t O he r i n d s a c o v t l u i n h w c t h e o n s i l u . i F g a 3 1 4 y b n ig u c t r o d e o r t w o u s n t i l e a r d t l i u s n i F g 3 .1 5 n e O t r a e d r i o uc nt i n t he a l i c ndr y t j i n o f a t l i on l e i d i ob t l y m i s oduc t e r i n i n t he t r f s i l c y

v 1 , ωδ )

( 2

F

p

v 1 v , 2 v , 3 , ω, β ωδ )

(

2 F

v 1 , ωδ )

(

0 F

T

2

3

M N

1 2

2

l

r

7

( v 1 , ωδ )

)

F

r

5

2 3

G2

R S

2

2

G1

M (

7

5 0

M

R

2 0

r

P R-P . ( g 3i F 1 4 b )

( v 1 , ωδ )

G1

r

PR

( v 1 v , 2 , ωδ )

G1

S

26

G1

)

R

(

e o p h t f a r c l smi n F . g 3 1 4

0 2

k

25

. T he s t l a t e ol r u v j t s oi n f o he t

0 2

8

1 12 13 14 15 16 17 18 19 20 21 2 23 24

T F =0

. esl ax

a Te l b 3 . 4

10

a nd

=1 F

c rt u a l e p m s

c a n be o t n d a i r m f

0